Bachelor Thesis

Knot and Homology Theory in Quantum Physics

performed at the

Institute for Analysis and Scientific Computing

at Vienna University of Technology

supervised by

Ao. Univ. Prof. Mag. rer. nat. Dr. phil. Wolfgang Herfort

by

Stefan Lindner

Matr.-No. 0925022

Vienna, 18.07.2012

Non scholae sed vitae discimus.

Abstract

In this thesis knot and homology theory are presented. Some of its aspects will be dis-cussed more detailed and finally hints to applications in modern quantum physics will beindicated.The first part deals with knot theory. Conventions of description will be discussed,particulary the description as polynomials. As one of the most important results theReidemeister’s theorem is proved. The Wilson-loop, whose expectation value is theJones-polynomial occurs in quantum field theory and closes the first part.The second part is about homology theory. Basic concepts such as the homology groupand simplicial complexes will be discussed and lead to the Jordan- Brouwer’s separation-theorem. These results also have their application in quantum theory, for example Kho-vanov homology.

Kurzfassung

Im Rahmen dieser Arbeit wird die Knoten- und Homologietheorie vorgestellt, einige As-pekte tiefergehend behandelt und Anwendungen in der modernen Quantenphysik angedeutet.Der erste Teil der Arbeit widmet sich der Knotentheorie. Es werden Darstellungskonven-tionen diskutiert und insbesondere auf die Darstellung als Polynome eingegangen. Alseines der wichtigsten Resultate wird der Satz von Reidemeister bewiesen. Der in An-wendungen der Quantenfeldtheorie auftretende Wilson-Operator, dessen Erwartungswertdas Jones-Polynom ist, schließt den ersten Teil.Der zweite Teil der Arbeit ist der Homologietheorie gewidmet. Grundbegriffe wie dieHomologiegruppe und Simplizialkomplexe werden diskutiert und fuhren zum Jordan-Brouwer’schen Separationssatz. Diese Resultate finden ebenfalls Anwendung in derQuantentheorie, zum Beispiel Khovanov-Homologie.

Abregee

Dans ce travail, la theorie des nœuds et la theorie de l’homologie sont presentees. Certainsaspects sont discutes de maniere plus detaillee et les applications de la physique quantiquemoderne sont demontres.La premiere partie du travail est consacree a la theorie des nœuds. Les conventionsde la representation des nœuds sont discutees, en particulier la representation avec despolynomes. Puis un des resultats les plus importants, le theoreme de Reidemeister,est prouve. L’operateur de Wilson avec les applications dans la theorie quantique deschamps, que le polynome de Jones est la valeur attendue, est la fin de la premiere partie.La deuxieme partie est consacree a la theorie d’homologie. Les concepts de base tels que legroupe d’homologie et des complexes simpliciaux seront discutes et aboutis au theoremede Jordan-Brouwer. Ces resultats sont aussi applicables dans la theorie quantique,par l’exemple dans l’homologie de Khovanov.

Acknowledgement

First of all I would like to thank my supervisor Wolfgang Herfort for his patientsupport and for having allowed much liberty in arranging the topics and details of mybachelor thesis. It was a great experience for me, to think about and discuss all thoseinteresting notions.My other professors I would also like to thank - for three years of very interesting bachelorstudies and support.Furthermore, I thank the librarians of the mathematics and physics library and the mainlibrary of the Vienna university of technology for their kind assistance in finding all thereferences.Also my collegues I would like to thank. It was a very pleasant atmosphere we had, whiledoing our bachelor studies and we surely will have during our master studies.And last but not least I would like to give many thanks to my family and specially tomy girlfriend - You always have been, are right now and will be in the future my mostimportant and valuable support!

Contents

1 Knot Theory 71.1 Canonic description of knots . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Knot diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.1 Over- and Underpasses . . . . . . . . . . . . . . . . . . . . . . . . 91.2.2 Alexander-Briggs-notation . . . . . . . . . . . . . . . . . . . 91.2.3 Description after Dowker . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Reidemeister moves and Reidemeister’s Theorem . . . . . . . . . . 121.3.1 Reidemeister moves . . . . . . . . . . . . . . . . . . . . . . . . 121.3.2 Reidemeister’s Theorem . . . . . . . . . . . . . . . . . . . . . . 141.3.3 Knot invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4 Knot-polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4.1 The L- polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4.2 The Jones polynomial . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5 The fundamental group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.5.2 Description of knots . . . . . . . . . . . . . . . . . . . . . . . . . 211.5.3 Fundamental group and Jones-polynomial . . . . . . . . . . . . . 22

1.6 Application in Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . 241.6.1 Physical Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.6.2 Knot theory applications . . . . . . . . . . . . . . . . . . . . . . . 25

2 Homology Theory 272.1 Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1.1 Simplicial complexes . . . . . . . . . . . . . . . . . . . . . . . . . 272.1.2 Homology group . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.1.3 Jordan - Brouwer separation theorem . . . . . . . . . . . . . . 29

2.2 Application in Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . 292.2.1 Khovanov homology . . . . . . . . . . . . . . . . . . . . . . . . . 29

5

Introduction and historical aspects

Different forms of knots have been known since long ago. They had technical applications,like in shipping, or weaving. More interesting and in particular aesthetically pleasing werethe mystic symbols in many cultures. Famous examples are the sign of the pythagorians,a pentagonal star, the David’s star, or the celtic trefoil knot, shown in the picture below.Wearing a ring or necklace with a trefoil knot, brings luck and safety over its owner dueto the beliefs of the Celts.First considerations about mathematical knots were done by Gauss in the 19th century.He developed strategies to compute the linking number of two knots. This number is afirst information about the grade of complexity of the linking of the considered knots.But Gauss found no further applications for knots. In the sixties of the same century,William Thomson, better known as Lord Kelvin, postulated that atoms are knotsin a mysterious substance called ”ether”. In the early 20th century, modern topology de-veloped, topologists like J.W.Alexander or Max Dehn began to consider knots froma topology point of view. That was the beginning of the success of knot theory. Furtherdevelopments led deeper and deeper into topology. One example is homology theory,which to get discussed later. Practical applications of knot invariants were hardly foundfor a long time, because calculations take much time. But when powerful computers weredeveloped, this problem got under control too. Nowadays there are many interesting ap-plications of knot theory and its further developments, for instance in quantum physics.A small selection of such applications will be discussed in this thesis.

Figure 1: Celtic trefoil knot necklace for my girlfriend

6

Chapter 1

Knot Theory

Knots will be interpreted as closed curves in R3. Later we will define polynomials to getan algebraic expression for a knot.

1.1 Canonic description of knots

We consider knots to be double point free closed polygons in euclidean space Rn:

Definition 1.1 Let γ be a path in Rn with γ : [a, b] → Rn, [a, b] ⊂ R. Let Z = ξi : i =0, . . . n(Z) be a decomposition of γ([a, b]). Further, let be ξ0 = ξn(Z). Then the sequence

Pi =−−−−→ξi, ξi+1 0 ≤ i ≤ n(Z)− 1 (1.1)

is called a polygon. Every segment Pi =−−−−→ξi, ξi+1 is called a strand.

The following figure shows a polygon in this notation, where n(Z) = 7: Graphically

Figure 1.1: Exemplaric knot

every polygon can be seen as a knot. But in order to be able to compare two knots we

7

intruduce certain eqivalence classes. We call polygons, which can be deformed one intoanother without passing through a self-intersection, the same knot. For giving a precisedefinition, we need the concept of deformation. Therefore we define two operations ∆and ∆′:

Definition 1.2 The operation ∆: Let−−→ξp, ξq, p < q be one strand of the polyhedron (Pi).

(Pi) has the endpoints ξp and ξq. Let−−−→ξpξp+1 and

−−−→ξp+1ξq be segments, which are not part

of the polyhedron. If the triangle area ξpξp+1ξq has no point in common (Pn) outside the

route−−→ξpξq,

−−→ξpξq is substituted with

−−−→ξpξp+1 ∪

−−−→ξp+1ξq.

The operation ∆′: ∆′ is the inverse operation of ∆. If the triangle area ξpξp+1ξq has no

point in common with P , except−−−→ξpξp+1∪

−−−→ξp+1ξq, then

−−−→ξpξp+1∪

−−−→ξp+1ξq is substituted by

−−→ξpξq.

Figure 1.2: Applying ∆ and ∆′ to a knot

Now we can give the definition of a knot:

Definition 1.3 A polygon (Pi)k, k ∈ N, which results from another polygon (Pn)0 byapplying a finite sequence of operations ∆ and ∆′ is called isotopic to the polygon (Pn)0.All polygons (Pn)i which are isotopic to (Pn)0 constitute an equivalence class of isotopicpolygons. Every such equivalence class [Pi] is called a knot.

Finally we define a special kind of knots:

Definition 1.4 A knot [Pi] which is isotopic to a triangle is called circle. Otherwise itis called knotted.

8

1.2 Knot diagrams

1.2.1 Over- and Underpasses

Before we come to the definition of the three Reidemeister-moves, we need two simple,but powerful terms. Therefore we have to provide the knot with an orientation, choosinga starting point ξ0. It will be indicated by arrows. To be able to discribe the knot in thisway, we need a projection π into a plane. There we get crossing points of the images oftwo strands. The projection is chosen in a way, that no double crossings occur:

Definition 1.5 Let [Pi] be a knot and π be a projection. Further let Pp and Pq be twostrands of the knot. Two points ζk and ζl on this strands in R2 with the property

ζk = π(Pp) = π(Pq) = ζl (1.2)

are called crossing points. A pair of crossing points at the same place that arise fromdifferent strands we call a crossing. The projection π is chosen in a way, that there donot occur any double-crossings, so

@Pr : π(Pp) = π(Pq) = π(Pr) (1.3)

We now define over- and underpass :

Definition 1.6 Considering two strands of a knot [Pi], Pp and Pq which cross each otherin a crossing point ζk, k ∈ N, k < ∞, following the knot’s orientation, there are twopossibilities to cross. We define the characteristics ε as follows:

ε =

−1 if Pp passes above Pq, which is called overpass

+1 if Pp passes below Pq, which is called underpass(1.4)

Figure 1.3: over- and underpass

1.2.2 Alexander-Briggs-notation

The Alexander-Briggs-notation characterizes a knot by the number of crossings itself,the crossing number:

9

Definition 1.7 Let π : R3 → R2 be a projection, (Pi) a representative of a knot and ζand η be crossing points of (Pi). The cardinality of the set

C = ζ : ∃η : π(ζ) = π(η), ζ 6= η (1.5)

is called crossing number of the representative (Pi). Every knot with crossing number 0is called unknot.

The minimal crossing number can be shown to be a knot invariant:

Lemma 1.8 Let [Pi] be a knot, (Pi)k be a representative and π a projection. If theminimal crossing number of (Pi) is C, then the minimal crossing number of every otherrepresantative (Pi)j, j 6= k is C and we can write C = C([Pi]).

Figure 1.4: Knottable, [wiki]

The more crossings a knot has, the more complex it is and the more representativesits equivalence class has. Figure 1.4 shows the possible knots for a minimal crossingnumber from 0 to 7. The knot with two crossings is equivalent to the unknot; it is notshown. The knots are given in Alexander-Briggs-notation. This is the simpliest no-tation for knots: It categorizes them only by their minimal crossing number, as seen infigure 1.4. The index is given arbitrarily and has no special mathematical meaning. Itonly shows, for instance, that there are two different knots with a crossing number offive. The Alexander-Briggs-notation gives little information about the knot, but itwas the first systematic categorisation of knots.It is now interesting, how many different knots with given minimal crossing number canbe found. No analytical expression has been found yet to calculate this number for acrossing number n. Only an upper bound on the number of knots with crossing numberof n that grows exponentially with n is known. For low crossing numbers following datahave been calculated: 1

1[knot1, section 2.1., page 47]

10

crossing number number of prime knots3 14 15 26 37 78 219 4910 16511 55212 217613 998814 4697215 253293

1.2.3 Description after Dowker

The Dowker-notation is based on how two strands of a crossing behave. 2 First wewill describe this notation for alternating knots and extend later on the notation to non-alternating ones.

Definition 1.9 A knot is called alternating, if the the crossings of a knot in a givenprojection π forms an alternating sequence of over- and underpasses.Otherwise the knot is called non-alternating.

The first step is again to choose an orientation of the knot and a first crossing point. Thenwe follow the knot along the chosen orientation and continue numbering the crossingpoints until we return to the first one. This process equips every crossing with twonumbers, called its Dowker-pair:

Definition 1.10 Let [Pi] be a knot, π a projection and C the crossing number of the knotin this projection. Then there is a sequence (ζk)

2Ck=0 of crossing points. Further let (Pq)

2Cq=0

be the sequence of the corresponding crossings. A Dowker-pair is a pair of indices ofcrossing points:

D = (a, b) ∈ N2 : ζa = ζb : π−1(ζa) ∈ Pp ∧ π−1(ζb) ∈ Pq, p 6= q (1.6)

D is the set of Dowker-pairs and C := |D| is the crossing number.

Every Dowker-pair has the following property:

Lemma 1.11 Every Dowker-pair (a, b) consists of an even and an odd number.

Proof. Let a be even, so a = 2x. For odd a the proof is analogous. There are twocases:

• C even: Then C = 2y, for y ∈ N. So the last crossing we give its first number is2y − 2x = 2(x − y). This is an even number. So the crossing with number a gets2(x− y) + 1 as its second number, which is odd.

2[knot1, section 2.2., page 49]

11

• C odd: Then C = 2y−1. So the last crossing getting its first number is 2y−1−2x =2(x− y)− 1, an odd number. So crossing number a gets 2(x− y)− 1 + 1 = 2(x− y)as second number, which is even.

Now we can produce a table of crossing numbers. Writing the odd numbers in ascendingorder, using lemma 1.11, yields a sequence of the corresponding even numbers, describingthe knot.Now we extend the notation to non-alternating knots. It is analogous to the methodfor alternating knots, except one additional step: Considering the knot’s orientation, inevery overpass the even number of the Dowker-pair gets a negative sign. Otherwise thesign is positive. Here is an example:

Figure 1.5: A knot with Dowker-notation (−6, 12,−2,−8, 4, 10), [wiki]

1.3 Reidemeister moves and Reidemeister’s Theo-

rem

We come to the main result of the first chapter. Reidemeister proved that only threeoperations, called the Reidemeister-moves Ω1,Ω2,Ω3, are sufficient to get all projec-tions of a knot. The Reidemeister moves are only locally defined operations, so we willconsider single strands of a given knot.

1.3.1 Reidemeister moves

Recall the two basic deformation operations ∆ and ∆′ from section 1.2. These operationsare yet defined for polygons. To define them for differentiable curves, we need threeelementary operations, called Reidemeister-moves.

12

Definition 1.12 Considering a single strand Pp, the first Reidemeister move Ω1 addsa loop to it. So increases the crossing number by 1. Using the crossing number operatordefined in 1.7 this can be written as

C(Ω1(π(Pp))) = C(π(Pp)) + 1 (1.7)

The inverse operator Ω−11 reverses the deformation:

C(Ω−11 (π(Zi))) = C(π(Zi))− 1 (1.8)

Figure 1.6: First Reidemeister-move Ω1, [wiki]

The second Reidemeister-move Ω2 turns non-overlapping strands into overlapping ones:

Figure 1.7: Second Reidemeister-move Ω2, [wiki]

Definition 1.13 The second Reidemeister move turns two strands Pp and Pq into aninterlacing with two crossings. The crossing number of the knot representative increasesby 2. Using the crossing number operator defined in 1.7 this can be written as

C(Ω2(π(Zi) ∪ π(Zj))) = C(π(Zi)) + C(π(Zj)) + 2 (1.9)

The inverse operator Ω−11 reverses the deformation:

C(Ω−12 (π(Zi) ∪ π(Zj))) = C(Ω2(π(Zi) ∪ π(Zj)))− 2 (1.10)

13

For two unknotted strands the result of applying Ω2 on this system is shown in figure 1.7.The last of the three Reidemeister-moves concerns three strands of the knot forminga triangle. Applying the operator Ω3, the strand on the very upper side gets moved overthe crossing of the other two strands:

Definition 1.14 Considering three strands Pi, Pj and Pk having exactly 3 crossings aredeformed as follows: Let Pk be the strand at the very up side and Pi the one at the verydown side. Then the caracteristics ε of the crossing points ζp, p ∈ i1, i2, j1, j2, k1, k2change under Ω3 without restriction of generality by

ε

Ω3(ζi1)Ω3(ζi2)Ω3(ζj1)Ω3(ζj2)Ω3(ζk1)Ω3(ζk2)

=

1 0 0 0 0 00 1 0 0 0 00 0 −1 0 0 00 0 0 −1 0 00 0 0 0 1 00 0 0 0 0 1

· ε

ζi1ζi2ζj1ζj2ζk1ζk2

(1.11)

The inverse operator Ω−13 reverses the deformation. This amounts to matrix multiplica-

tion with the same matrix because the matrix is involutoric.

Figure 1.8: Third Reidemeister-move Ω3, [wiki]

1.3.2 Reidemeister’s Theorem

The three operations Ωi, i ∈ 1, 2, 3 are sufficient to deform a representative of a knot[Pi] into any other. This is the assertion of Reidemeister’s theorem 3:

Theorem 1.15 (Reidemeister) With the operations Ω1, Ω2 and Ω3 every change ina projection π of a knot A[(Pn)] which results from a knot deformation ∆ or ∆′ can bedescribed. These operations , including their inverse Ω−1

1 , Ω−12 and Ω−1

3 are sufficient.

Proof. Without restriction of generality we consider a special polygon (Pn). In sec-tion 1.1 it was shown, that the polygons set up an equivalence class, so the results weget for one special polygon apply to the whole knot. Furthermore we consider only one

3[knot2, section 3, page 8]

14

direction, the other is exactly the inverse process, where we use ∆′ and Ω−11 ,Ω−1

2 ,Ω−13

instead of ∆ and Ω1,Ω2,Ω3.

We consider the deformation ∆. Applying ∆ to the knot, a strand−−→ξpξq gets substituted by

−−−→ξpξp+1∪

−−−→ξp+1ξq. The points π(ξp), π(ξp+1) and π(ξq) are not collinear. We can assume, that

the projections of the initial polygon and the deformed one, are both regular. So there areonly double points. The triangle π(ξp)π(ξp+1)π(ξq) has only finitely many double pointsinside and on its edges. Now we can decompose the triangle into straight lines parallel

to−−→ξpξq and

−−−→ξpξp+1 with the following properties: The straight lines confine triangles and

parallelograms such that the projections the corresponding triangles and parallelogramscontain at most one double point. Further, they get crossed by four strands π(Pi) of thepolygon’s projection if there is a double point inside, or it gets crossed by at most onestrand if there is none.−−→ξpξq we can convert into

−−−→ξpξp+1 ∪

−−−→ξp+1ξq by applying a finite sequence of Reidemeister-

moves Ω1 and Ω2 and their inverses. These triangles and parallelograms dissolve step bystep.The only problem left in a general position are singularities in the projections (we as-sumed the projections to be regular). But those can be attributed to regular projections:Let π1 be a singular projection. The Reidemeister-moves are not dependend on theprojection, so we can get the solution by changing to another, regular projection π2 usingthe chain:

π1(〈©〉) = π2 [F n] π2 π−11 π1(〈L〉)

where [F n] is a chain of n Reidemeister-moves and L is our polygon.

1.3.3 Knot invariants

Knot invariants are expressions, which do not change under deformations ∆ and ∆′ or theReidemeister-moves. It has to be mentioned at this point, that finding knot invariantsis a central problem in knot theory. A complete set of knot invariants has not been foundyet, it is not even known, if there is a finite number of knot invariants. First we considerthe unknotting number.

Definition 1.16 Let [Pi] be a knot. Let K0 denote the unknot. The function t : R3 →R3 : [Pi] 7→ [Qj] changes the crossings of the knot [Pi], so that one overpass gets turnedinto an underpass or one underpass gets turned into an overpass. So we get a new knot[Qi]. Further let w ∈ N. If there exists a projection π : R3 → R2, so that

π tw([Pi]) = π(K0) (1.12)

then w is called the unknotting number of the knot [Pi].

The unknotting number indicates, how many crossings have to be changed to get theunknot.The second knot invariant to be defined is the overpass number:

Definition 1.17 Let [Pi] be a knot in R3, π a projection and b an overpass of the knot.A maximal overpass b([Pi]) is an overpass, which cannot be extended to another crossing.

15

The overpassing number O([Pi]) is defined as

O([Pi]) = minbb([Pi]) = sup

bb([Pi]) (1.13)

The last knot invariant which we discuss now is the self-intersection number:

Definition 1.18 Let [Pi] ∈ K be a knot, (πi)i∈N a sequence of projections R3 → R2 andC : K → N the function, which defines the crossing number. Then the number

C([Pi]) = mini∈NC(πi([Pi])) (1.14)

is called self-intersection number.

1.4 Knot-polynomials

The ideas of Alexander, Jones and others to describe knots by means of polynomialswas very successful.

1.4.1 The L- polynomial

The L−polynomial has been introduced by K. Reidemeister in need of finding knotinvariants. 4 Consider a knot [Pi]. Every crossing has an unique Dowker-pair (ζi, ζj).Let λ : N → N : i 7→ j be the bijective function described by the Dowker-pairs, soj = λ(i). Now we define a matrix lij(x) ∈ (K[X])2C×2C , where C is the crossing numberof the considered knot. The rows of this matrix describe the crossings (ζi, ζj) and thecolumns describe the corresponding strands Pi and Pj.In the row of (ζi, ζj) with εi = 1 and j = λ(i) 6= i or i+ 1 we write i+ 1. In the columnof Pi we write X, −1 in the one of Pi+1 and 1−X we write in the column of Pλ(i). Forε = 1 and λ(i) = i we write 1 in the column of Pi and −1 in the one of Pi+1 and for εi = 1and λ(i) = i+ 1 we write X in the column of Pi and −X in the one of Pi+1. Further forε = −1 and λ(i) 6= i or i + 1 we write 1 in the column of Pi, −X in the one of Pi+1 andX − 1 in the one of Pλ(i). For ε = −1 and λ(i) = i we write X in the column of Pi and−X in the one of Pi+1 and for εi = −1 and λ(i) = i + 1 we write 1 in the column of Piand −1 in the one of Pi+1. The other entries are set zero.Now we define an equivalence class of matrices (lik(x)) as follows:

(lik(x)) ∼ (mik(x))⇔ (lik(x)) is similar to (mik(x))

up to adding or removing zero-rows and -columns

This yields L-equivalent matrices. It can be shown, that the L-equivalence class of a ma-trix (lik(x)) is a knot invariant. All representatives of a knot yield the same determinant.This is the definition of the L-polynomial:

Definition 1.19 Let π([Pi]) be a projection of a knot and lik(x) the L-matrix of one ofits representatives. Then the L-polynomial of the [Pi] is defined as

L(X) = det (lik(x)) (1.15)

16

Figure 1.9: trefoil knot

As an example we calculate the L-polynomial of the trefoil knot. According to figure 1.8,j = λ(i) = i+ 3. We now can determine the matrix lij of the trefoil knot T :

lij[T ](x) =

1 −x 0 x− 1 0 00 x −1 0 −x+ 1 00 0 x −1 0 −x+ 1

x− 1 0 0 1 −x 00 −x+ 1 −1 0 x −1−1 0 −x −1 0 x

(1.16)

The L-polynomial of the trefoil knot is the determinant of this matrix:

det lij[T ](x) = −2x6 + 7x5 − 11x4 + 11x3 − 4x2 − x (1.17)

1.4.2 The Jones polynomial5 The Jones-polynomial relies on the concept of the bracket polynomial.6 Like the L-polynomial the bracket- and Jones-polynomial uses interlacings.First we require that the bracket polynomial of the unknot, which is the unit element inthe space of knots, is the unit element of the polynomial algebra 1:

〈©〉 = 1 (1.18)

There are shortcuts for some typical kinds of interlacings as shown below7.An interlacing in the considered projection we divide into two new projections of theinterlacing. Each of them has now less crossings than the initial interlacing. This processis reversible: Putting the two interlacings together yields the initial one; we require that

4[knot2, section 14, page 37]5[knot1, section6.1, pages 157-162]6[knot1, section6.1, pages 157-162]7[knot3]

17

Figure 1.10: Interlacing types

the polynomials reflect this behavior. This can be achieved by defining this operation asa linear combination with coefficients a, b. The polynomial of the projections of the newinterlacings has to be eqal to the polynomial of the projection of the initial interlacing:

〈A〉 = a〈B〉+ b〈B′〉 (1.19)

〈A′〉 = a〈B′〉+ b〈B〉 (1.20)

The difference between these two equations is the direction of the projection. The direc-tions are perpendicular. So we see, that those two projections are equivalent from thepolynomial point of view. There is the same structure. For adding an interlacing withan unknot we define

〈L ∪©〉 = c〈L〉 (1.21)

because it has to be possible to decompose again the result of this addition. L denotesthe initial interlacing and c is another scalar variable.In order to make the polynomial independent from the projection π, it has especially tobe an invariant when changing the interlacing. Every change of the projection can be seenas a multiple interlacing of the knot. To precisize this, we use the Reidemeister-movesas follows:

〈Ω1(L)〉 = 〈L〉 (1.22)

〈Ω2(L)〉 = 〈L〉 (1.23)

〈Ω3(L)〉 = 〈L〉 (1.24)

We do this in order to turn 〈L〉 into a knot invariant. First we consider the operationΩ2. This means that we require

〈B′′〉 = 〈B〉

18

As a consequence we have b = a−1. So the coefficient of 〈B〉 is 1 and we have reducedthe number of variables by 1. Considering condition (1.21) we see that

a2 + c+ a−2 = 0 (1.25)

So we can eliminate one more variable by setting c = −a−2 − a2. Now we can write thethree requirements for the bracket polynomial (1.18), (1.19) and (1.21) using the singlevariable a. Therefore we need the following algebraic transformation of interlacing B′′:

〈B′′〉 = a〈C0〉+ b〈D〉 =

= a(a〈C1〉+ b〈C2〉) + b(a〈E〉+ b〈C ′1〉) =

= a(a〈B′〉+ bc〈B′〉) + b(a〈B〉+ b〈B′〉) =

= (a2 + abc+ b2)〈B′〉+ ba〈B〉 = 〈B〉

Now the requirements (1.18), (1.19) and (1.21) become:

(1.18) : 〈©〉 = 1 (1.26)

(1.19) : 〈A〉 = a〈B〉+ a−1〈B′〉 ⇔ 〈A′〉 = a〈B′〉+ a−1〈B〉 (1.27)

(1.21) : 〈L ∪©〉 = (−a−2 − a2)〈L〉 (1.28)

The next step is to consider the third Reidemeister-move Ω3:

Ω3(F1) = F ′1

That has no influence on the bracket polynomial:

〈F1〉 = a〈F2〉+ a−1〈F3〉 = a〈F ′2〉+ a−1〈F ′3〉 = 〈F ′1〉

As last step we consider the first Reidemeister-move Ω1:

Ω1(H0) = H4

There is a problem with this deformation: It depends on the projection of the intersection:

〈H0〉 = a〈H1〉+ a−1〈H2〉 =

= a(−a−2 − a2)〈H3〉+ a−1〈H3〉 =

= −a3〈H4〉〈H ′0〉 = a〈H2〉+ a−1〈H1〉 =

= a〈H4〉+ a−1(−a−2 − a2)〈H4〉 =

= −a−3〈H4〉

So we would get

〈H0〉 = 〈H ′0〉−1

This can be resolved by orienting the knot and using the characteristics ε of crossingpoints. We define a new knot operator, the winding-number:

19

Definition 1.20 Let [Pi] be a subset of a knot (e.g. an intersection) and π : R3 → R2 aprojection. Considering π([Pi]), the knot has crossings (ζi, ζλ(i)). The operator

ω([Pi]) :=

C(π([Pi])∑k=1

ε((ζk, ζλ(k))) (1.29)

is called winding-number.

One can show that ω is invariant if Ω2 and Ω3 is applied to the interlacing.

Definition 1.21 X-polynomial:

X(L) =(−a3

)−ω(L) 〈L〉 (1.30)

Ω2 and Ω3 have no influence on ω(L) and 〈L〉, hence only sign can switch. Next we usethe following result, which can be directly seen from the definition of Ω1 and ω:

ω(Ω1(L)) = ω(L)± 1 (1.31)

Now we can analyse the influence of Ω1 on X(L):

X(Ω1(L)) = (−a3)−ω(Ω1(L))〈Ω1(L)〉 =

= (−a3)−(ω(L)+1)〈Ω1(L)〉 =

= (−a3)−(ω(L)+1)((−a)3〈L〉

)=

= (−a3)−ω(L)〈L〉 =

= X(L)

As an example we calculate X(T ) for the trefoil knot T . Using

〈G2〉 = a〈G3〉+ a−1〈G4〉 =

= a(−a1−2− a−2) + a−1 =

= −a−3

and 〈G1〉 = −a3 we get

〈T 〉 = a〈G0〉+ a−1(−a−3〈G2〉) =

= a(a〈G1〉+ a−1〈G2〉) + a−1(−a−3(−a−3)) =

= a(a(−a3〈©〉) + a−1(−a−3〈©〉)) + a−7 =

= a(−a4 − a−4) + a−7 =

= −a−5 − a−3 + a−7

The definition of the Jones-polynomial differs from the X-polynomial only by a variabletransformation:

Definition 1.22 Let L be an intersection and t ∈ R a variable. Then the Jones-polynomial V (L) is defined as

VL(t) :=(−t−

34

)−ω(L)

〈L〉(a(t)) (1.32)

With a(t) = t−34 . The formula symbol ”V ” should remind of ”Vaughan Jones”.

20

1.5 The fundamental group

The fundamental group is another knot invariant. First we define the fundamental groupfor an arbitrary topological space and iterprete it then for knots.

1.5.1 Definition8

Definition 1.23 Let X and Y be topological spaces. A family of functions ht : X → Ywith the parameter t ∈ [0, 1] is called homotopy, if the function

H :

X × [0, 1]→ Y

(x, t) 7→ ht(x)(1.33)

is continous with respect to the product topology on X× [0, 1]. Two functions f and g arecalled homotopic, if there exists a homotopy ht : X → Y with h0 = f and h1 = g Thenwe write f ∼ g. ∼ is an equivalence relation and the equivalence class

[f ] = g : X → Y : g ∼ f (1.34)

is called the homotopy class of f .

Definition 1.24 Let X be a topological space and fix x0 ∈ X. The pair (X, x0) iscalled space with base-point. Let (Y, y0) be another space with base-point. A functionf : (X, x0) → (Y, y0) or a homotopy ht : (X, x0) → (Y, y0) is called relative to base-points.

Considering paths in X we can now define the fundamental group:

Definition 1.25 Let (X, x0) be a topological space with base-point. Then the set

π1(X, x0) = [ω] : ω : [0, 1]→ X finite path with base point x0 (1.35)

is called fundamental group of X with base-point x0. It is a group with the multiplication· : ·([p], [q]) 7→ [p · q]. Its neutral element 1x0 is the homotopy class of the zero-homotopicpaths. The inverse element of [ω] is [ω−1], where ω−1 is the path traversed in oppositedirection.

1.5.2 Description of knots

A homotopy class we can depict as a family of pathes which the same endpoints. Theydeform into each other without self-intersection, as the following figure shows. This issimilar to the idea of knots: There we considered different polygons which transfer intoeach other by the Reidemeister-moves Ω1,Ω2 and Ω3. Now we have a set of equivalenceclasses of paths. It is a very similar concept with the advantage that we can use thepowerful instruments of algebraic topology for describing these ”new” knots. They are

8[algtop, section 5.1, page 103]

21

Figure 1.11: Homotopic functions relative to x and y, [wiki]

new in that sense, that we have to consider the complementary space of the knot and notthe knot itself. On these objects homotopic functions can be found, which correspond tothe Reidemeister moves. There is only one disadvantage: In previous considerationsthere was no restriction for the knots concerning any special points. Now we have aspecial point, the distinguished point x0. In many applications this is no problem andthe topological advantages weigh out, so the topological view on knots is the second one,next to the description by polynomials, which is already used in modern studies.

1.5.3 Fundamental group and Jones-polynomial

There is a connection between the fundamental group of a knot and its Jones-polynomial.The first step is to consider the knot as a closed set of points in R3. The knots’ com-plement is infinite and open. In the following figure this is indicated with the trefoilknot where we replace R3 by a torus with a disk glued along its inner radius. Our ge-ometric interpretation is suitable for any torus-knot. The next step is to intersect the

Figure 1.12: Applying the complement operator to the trefoil knot

torus with a plane as indicated in figure 1.12. This happens in this way, that there areno ”lost” strands left - every strand has to have an endpoint on the plane. To makethis special kind of intersection more clear, it is shown again from another point of viewin figure 1.12, where b denotes a not more detailed defined function which applies thedescribed intersection to the knot and c the complement operator. h is a homotopy whichwe will discuss in the next step. After embedding the knot and interecting the torus, the

22

Figure 1.13: Cutting, embedding and homotopy

torus gets bended to a cylinder as shown in figure 1.13. Then a homotopy h is appliedwhich untwists the embedded knot. The knot is actually considered as a ”hole” in thecomplementary space. It means that no self-intersection during deformations applies tothe knot. After performing all these transformations to the knot, its fundamental groupcan be seen: Choosing a base point d in the complementary space, the fundamental grouparises from n closed loops with common base point d. Here n is the number of strands– it is two in figure 1.13. Every loop surrounds a single strand: Applying a projection

Figure 1.14: fundamental and free group

π on the cylinder yields a deformation as indicated in the figure. The space obtainedby glueing copies of the cylinder allows shift – with quotient our torus. Thus the infi-nite cyclic group Z appears as a quotient of the knot group G – with kernel Fn the freegroup generated by the n strands in the cylinder. Hence Fn contains G′, the commutatorsubgroup of G, since Z is abelian. Actually it turns out that Fn = G′. Now G actson its commutator subgroup G′ by conjugation and so induces an action of G/G′ uponG′/G′′. So with t a generator of G/G′ we find the group ring Z[t, t−] to act on the finitelygenerated Z[t]-module G′/G′′. Expanding the determinant of t yields the L-polynomial– the latter transforms into the Jones-polynomial.

23

1.6 Application in Quantum Theory

Knot theory has many applications in quantum theory. The most prominent one is stringtheory, which uses higher dimensional knots.Here we will discuss the Wilson-loop, or Wilson-operator, named after KennethWilson. At first, a few physical principles will be given.

1.6.1 Physical Basics

Particles as wave functions9 One of the main basic assumptions of quantum theory is the view of particles as wavefunctions. We assume the particle beeing in a location-dependent potential V (x). Mostlyelectrons are considered and for them V is the Coulomb-potential. The influence ofthe weak force is neglible in general. Now the motion equation of the electron has tobe etablished. Generalizing classical mechanics, developed by Newton, in theoreticalmechanics a certain formalism was developed. It is the Lagrange- and Hamilton- for-malism, where the whole information of the particles motion is described by a differentialoperator, the Hamiltonian. For the harmonic oscillator it is:

H = − ~2

2m

∂2

∂x2(1.36)

The location of the particle is described by a wave-function ψ, which satisfies the Schrodinger-equation

Eψ = Hψ (1.37)

Here E is the energy-eigenvalue of the particle.

Observables as operatorsEvery observable is regarded as an operator. Scalars become replaced by functions ina function space, e.g. in L2. This leads to another important instrument of quantummechanics:

Results of measurement as expectation value of the corresponding operatorAll values that result from measurements correspond with one of the eigenvalues of thecorresponding operator to the observable. The most probable value is the expectationvalue of that operator. In classical mechanics, several measurements are performed andthe most propable value is the average of the data, i.e.

x =1

n

n∑k=1

xk

If now the equation is multiplied with n, we get a sum without any scalar factors. Because,considering physical measurements, we are interested in the absolute value of deviation,

9[quantum]

24

we substitute xk by its squares x2k. So we get a new expression

X =n∑k=1

x2k

which also can be seen as a measure of total deviation. This expression is nothing elsethan the scalar product of the vector −→x of the measurement data with itself.We proceed similarly on the function space L2, using the L2-scalarproduct

〈f |g〉 =

∫Ω

f · g∗ dλ(x)

The expectation value of an observable O as follows:

〈O〉 = 〈ψ|Oψ∗〉

1.6.2 Knot theory applications

The concepts presented below are quite complex and we can only indicate them.

Wilson-loop10. The Wilson-loop is an interlacing, defined as

WC = Tr

(exp

(i

∮C

Aµdxµ

))(1.38)

Here Tr denotes the trace of the operator, C is a curve in 4-space and A a vector po-tential. The Wilson-loop is used for instance in theoretical nuclear physics to describequark states by quantum chromo dynamics. Its expectation value turns out to be theJones-polynomial of the knot C in R4.

Theoretical solid state physicsIn recent years amorph structures (glass) became more and more interesting. Theirdescription leads to multiply knotted lines with no short-range order. Applying the un-knotting results, gives a method for describing degrees of freedom of thermal movements,the basis estimating entropy and heat capacity.

Quartary structure of proteinsProtein molecules have a very complex wide-range order, the so called quartary struc-ture. The protein strands, consisting of amino acids fold themselfes as a result of theCoulomb-attraction between the amino and carboxyl groups. Considering the proteinstrands as knots with the charge relations as side conditions knot theory could help tounderstand better ”protein folding”.

Spin networks and discrete geometry in quantum gravity11 For unification of relativity theory and quantum physics quantisation of gravity is

10[phys1], [phys2]11[rovelli]

25

Figure 1.15: Structure of a SiO2-glass, [wiki]

Figure 1.16: Quartary structure of haemoglobine, [wiki]

needed. This has not been developed far enough yet. One promising idea was to lookfor a quantisation of the space, a kind of discrete geometry, instead of a direct quantisa-tion of gravity. This is motivated by Einstein’s general relativity theory, where spaceand time become geometrical objects. The idea of Rovelli and others was now to findquantisation conditions. This has been done by constrained systems. Two of them, theGauss-constrained and the diffeomorphism constrained are understood quite well. Herea base of operators set up the spin networks. They describe a kind of discrete geome-try, which can be understood as knots. The theory of unknotting has its applications inconsidering the last constraint condition, the Hamilton constraint. So the Hamiltonianof usual quantum systems becomes a constraint condition too in discrete geometries. Itdescribes an area operator, whose eigenvalues describe the behavoir of the spin networksin a crossing point. If the area operator is applied to a multiple crossing, its spectrumchanges. This is known as recoupling theory and was developed by C.Rovelli.

26

Chapter 2

Homology Theory

The presented concepts belong to algebraic topology.

2.1 Homology

2.1.1 Simplicial complexes1 In the last chapter we considered knots as 1-dimensional curves in 3-dimensional space.The idea of simplicial complexes is to consider higher dimensions. We consider only theRn and no more abstract spaces. There we have polyhedrons instead of polygons. Firstwe need the definition of a polyhedron:

Definition 2.1 Let x0, . . . , xq ∈ Rn be points. The set

σq =

x ∈ Rn : x =

q∑i=0

λixi with

q∑i=0

λi = 1, λ0, . . . , λq > 0

∈ Rn (2.1)

is called the open q-simplex with corners x0, . . . , xq. q = dimσq is called the dimensionof the q-simplex. The set

σq =

x ∈ Rn : x =

q∑i=0

λixi with

q∑i=0

λi = 1, λ0, . . . , λq ≥ 0

∈ Rn (2.2)

is called the closed q-simplex with corners x0, . . . , xq.

When xi = ei, so considering the standard base of Rq+1, then σq is the q-dimensionalstandard simplex in Rq+1:

Definition 2.2 Let (ei)0≤i≤q be the standard base of Rq+1. The closed simplex with cor-ners e1, . . . eq is called the q-dimensional standard-simplex ∆q:

∆q =

x ∈ Rq+1 : x =

q+1∑i=0

λiei with 0 ≤ λi ≤ 1 and

q+1∑i=0

λi = 1

(2.3)

1[algtop, section 3.1., page 70-72]

27

Before we can define simplicial complexes we need an order relation on the set of sim-plexes:

Definition 2.3 Let σ and τ be simplexes in Rn. We call τ a face of σ and write τ ≤ σif the corners of τ are corners of σ too. If τ ≤ σ and τ 6= σ, τ is an actual face of σ andwe write τ < σ.

Now we can give the definition of simplicial complexes:

Definition 2.4 A simplicial complex K is a set of simplexes in Rn with the followingproperties:

1. σ ∈ K and τ < σ ⇒ τ ∈ K

2. σ, τ ∈ K and σ 6= τ ⇒ σ ∩ τ = ∅

The 0-simplexes of K are called corners of K and the 1-simplexes are called edges of K.The dimension of K is defined as

dimK = maxσ∈K

(dimσ) ∈ N ∪ ∞ (2.4)

Definition 2.5 Let K be a simplicial complex in Rn. The subset

|K| =⋃σ∈K

σ ⊂ Rn (2.5)

equipped with the weak topology is called the topological space of the simplicial complexK.

2.1.2 Homology group2 We can use the concept of simplicial complexes now to define the homology group.

Definition 2.6 (Homology group) Let X be a topological space. A singular q-simplexis a continous function σq : ∆q → X. The qth singular chaingroup Sq(X) of X is the freeabelian group generated by the q-simplexes in X. Its elements are called singular q-chainsin X. For q < 0 we define Sq(X) = 0. For q ≥ 1 the boundary operator is defined as thefollowing homomorphism:

∂q : Sq(X)→ Sq−1(X), σ 7→q+1∑i=1

(−1)i(σ δq−1,i) (2.6)

For q ≤ 0 we set ∂q = 0. The system S(X) = (Sq(X), ∂q)q∈Z is a chain complex:

S(X) : . . .∂−→ Sq+1(X)

∂−→ Sq(X)∂−→ Sq−1(X)

∂−→ . . .∂−→ S0(X)

∂−→ 0∂−→ 0

∂−→ . . . (2.7)

It is called the singular chain complex of X.The corresponding groups Hq(S(X)) are called(absolute) singular homology groups of X. We write them as Hq(X) := Hq(S(X)) andset them 0 for q < 0.

2[algtop, section 9.1., page 216]

28

The homology group shares some properties with the fundamental group defined in sec-tion 1.5.

Theorem 2.7 The function h1 : π1(X, x0)→ H1(X) is a natural homomorphism retard-ing continuous functions f : (X, x0) → (Y, y0). If X is a coherent CW-space 3, h1 issurjective and the kernel of h1 is the commutator-subgroup K of π1(X, x0).

We only indicate the proof:

Proof. The first step is to show, that h1 is a homomorphism. This is done byconsidering two functions φ, ψ : (S1, 1)→ (X, x0) and checking the properties of a homo-morphism. Afterwards we use the fact, that X is a CW-space 4. So we can consider thedecomposition of X which is given by its cells. They are circle-lines in this case, whichcan be deformed. Therefore we use a homotopy fk : S1 → X1. In the last step we canconstruct a commutative diagram, which proves the assertion.

2.1.3 Jordan - Brouwer separation theorem5 The Jordan-Brouwer separation theorem says, that the plane R2 is decomposed byevery single closed curve S ⊂ R2, or in other words, R2 \ S is not path-related. There isa generalisation of this theorem for higher dimensions.

Theorem 2.8 Let S ⊂ Rn be an r-sphere, where n ≥ 2 and 0 ≤ r ≤ n − 1. ThenHq(Rn \ S) = 0 except in the following two cases:

(a) For r < n− 1, Hq(Rn \ S) ∼= Z for q = 0, n− r − 1, n− 1 (2.8)

(b) For r = n− 1, H0(Rn \ S) ∼= Z⊕ Z and Hn−1(Rn \ S) ∼= Z. (2.9)

For n = 2 this is the Jordan’s curve theorem and for n > 2 it is the Brouwer’sseparation theorem.

2.2 Application in Quantum Theory

2.2.1 Khovanov homology6 Khovanov homology, has certain applications in quantum field theory. It is a powerfulgeneralistion of the Jones-polynomial defined in section 1.4.2. Further, the Khovanov-homology is a knot invariant and it can be shown, that it is the homology of a chaincomplex. Its construction is very similar to the one of the Jones polynomial. It uses atopological operator, the Khovanov bracket which has three properties, similar to thoseof the bracket polynomial. ∅ denotes an empty link, O an unlinked component and D a

3[algtop, section 9.8., page 245]4[algtop, section 9.8., page 245]5[algtop, section 11.7., page 301]6[phys1], [wiki]

29

crossing of the link L. V is a vector space. Further, F is an operator, which simplifies alink by forming a single complex out of a double complex.

(a) [∅] = 0 (2.10)

(b) [OD] = V ⊗ [D] (2.11)

(c) [D] = F (0→ [D0]→ [D1]1 → 0) (2.12)

The applications of the Jones polynomial and the Khovanov homology in theoreticalphysic reach from solid state physics over quantum field theory to string theory, see[phys1] or [knot3].

30

List of Figures

1 Celtic trefoil knot necklace for my girlfriend . . . . . . . . . . . . . . . . 6

1.1 Exemplaric knot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Applying ∆ and ∆′ to a knot . . . . . . . . . . . . . . . . . . . . . . . . 81.3 over- and underpass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Knottable, [wiki] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 A knot with Dowker-notation (−6, 12,−2,−8, 4, 10), [wiki] . . . . . . . 121.6 First Reidemeister-move Ω1, [wiki] . . . . . . . . . . . . . . . . . . . . 131.7 Second Reidemeister-move Ω2, [wiki] . . . . . . . . . . . . . . . . . . . 131.8 Third Reidemeister-move Ω3, [wiki] . . . . . . . . . . . . . . . . . . . . 141.9 trefoil knot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.10 Interlacing types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.11 Homotopic functions relative to x and y, [wiki] . . . . . . . . . . . . . . . 221.12 Applying the complement operator to the trefoil knot . . . . . . . . . . . 221.13 Cutting, embedding and homotopy . . . . . . . . . . . . . . . . . . . . . 231.14 fundamental and free group . . . . . . . . . . . . . . . . . . . . . . . . . 231.15 Structure of a SiO2-glass, [wiki] . . . . . . . . . . . . . . . . . . . . . . . 261.16 Quartary structure of haemoglobine, [wiki] . . . . . . . . . . . . . . . . . 26

31

Bibliography

[algtop] R. Stocker, H. Zieschang: Algebraische Topologie B.G. Teubner, Stuttgart1994.

[quantum] F. Schwabl:Quantenmechanik Springer, Heidelberg 2007.

[diffgeo] W. Kuhnel: Differentialgeometrie Vieweg + Teubner, Stuttgart 2010.

[linalg] H.Havlicek: Lineare Algebra fur technische Mathematiker Heldermann Verlag,Lemgo 2008.

[knot1] C.C. Adams: Das Knotenbuch: Einfuhrung in die mathematische Theorie derKnoten Spektrum Akad. Verlag, Heidelberg 1995.

[knot2] K. Reidemeister: Knotentheorie Springer, Berlin 1932.

[knot3] W.B. Raymond Lickorish: An Introduction to Knot Theory Springer, NewYork 1997

[phys1] E. Witten: Quantum Field Theory and the Jones polynomial Commun. Math.Phys. 121, 351-399 (1989)

[phys2] D. Bar-Natan: On Khovanov’s categorification of the Jones polynomial Algeb.Geom. Top. 2, 337-370 (2002)

[phys3] T. Fiedler: On the degree of the Jones Polynomial Topology 30/1, 1-8, 337-370(1991)

[analysis] M. Kaltenback: Analysis 2 fur technische Mathematik Skriptum zur Vor-lesung an der TU Wien, 2010

[rovelli] C. Rovelli: Quantum Gravity draft, november 30th

[wiki] Wikipedia, the free encyclopedia, Knot theory

32

Statutory declaration

Hereby I assure, that I wrote this present bachelor thesis with the title

Knot and Homology Theory in Quantum Physics

independently and that I used only, without exception, the indicated sources and aids.

Stefan LINDNER

33