JONES POLYNOMIALTy Callahan

Historical Background

Lord Kelvin thought that atoms could be knots

Mathematicians create table of knots

Organization sparks knot theory

Background

Knot A loop in R3

Unknot

Arc Portion of a knot

Diagram Depiction of a knot’s

projection to a plane

Diagram

OK NOT OK

Equivalence

Two knots are equivalent if there is an isotopy that deforms one link into the other

Isotopy Continuous deformation of ambient space Able to distort one into the other without breaking

Nothing more than trial and error can demonstrate equivalence Can mathematically distinguish between

nonequivalence

Figure 8 Knot

Orientation

Choice of the sense in which a knot can be traversed

Crossings

Orientation results in two possible crossings Right and Left

Jones Polynomial

Two Principles

1) Assign a value of 1 to any diagram representing an unknot

2) Skein Relation: Whenever three oriented diagrams differ at only one crossing, the Jones Polynomial is governed by the following equation

€

t−1R[t] − tL[t] = (t12 − t

−12)Q[t]

Ex. Trefoil Knots

1) Skein Relation for Right Trefoil

€

t−1R1[t] − t = (t12 − t

−12)Q1[t]

€

R1[t] = (t32 − t

12)Q1[t]+ t

2

2) Skein Relation for Link

€

t−1R2[t] − tL2[t] = (t12 − t

−12)

€

R2[t] = t2L2[t]+ t

32 − t

12

3) Skein Relation for Twisted Unknot

€

t−1 − t = (t12 − t

−12)Q3[t]

€

t−12 − t

32 = (t −1)Q3[t]

€

(t −1)(−t12 − t

−12 ) = (t −1)Q3[t]

€

Q3[t] = −t12 − t

−12

4) Substitute and Simplify

€

L2[t] =Q3[t] = −t12 − t

−12

€

R2[t] = t2(−t

12 − t

−12) + t

32 − t

12

€

R2[t] = −t52 − t

32 + t

32 − t

12

€

R2[t] = −t52 − t

12

4) Continued..

€

Q1[t] = R2[t] = −t52 − t

12

€

R1[t] = (t32 − t

12)(−t

52 − t

12) + t 2

€

R1[t] = −t4 − t 2 + t 3 + t + t 2

€

R1[t] = −t4 + t 3 + t

5) Compare to Left Trefoil

€

R1[t] = −t−4 + t−3 + t−1

€

R1[t] = −t4 + t 3 + tRight

Left

Conclusion

The Jones Polynomial of the Right Trefoil knot does not equal that of the Left Trefoil knot

The knots aren’t isotopes

“KNOT” EQUAL!!