SEIFERT

SURFACES

B Y RE B E C C A M

A R K O W I TZ

In 1930 the idea was first demonstrated by Frankl and Pontrjagin

In 1934 a German mathematician named Herbert Seifert came up with a simpler construction of the surface

HISTORY OF SEIFERT SURFACES

ORIENTABILITYAn Orientable Surface is one where if you start at one point on the surface then go all the way around and you will be in the exact same spot you started. Orientable Not Orientable

SEIFERT SURFACES PROPOSITIONA Seifert Surface of a knot (or link) is an orientable surface whose boundary coincides with that of the knot.

PROOF1.) Select a orientation for Knot K

2.) At each intersection of the knot eliminate the crossing.

SEIFERT CIRCLES

A Seifert Circle is a collection of unknotted simple closed curves

PROOF CONTINUED3.) Since we do not want the Seifert circles to intersect we will make them different heights rather than them all being in the same plane.

PROOF CONTINUED4.) We must connect the disks or seifert circles to each other with half twisted bands.

5.) In order to see that this surface is orientablewe need to color or paint each distinct side.

SEIFERT SURFACESNote, there might be more than one seifert surface for the same knot, because you can alter the projection of a knot and change the way the surface looks.

QUESTIONS OR COMMENTS