Spring 2019 - Math 191 - Knot Theory

Instructor: Michael Klug
Email: mrklug at berkeley dot edu
Meeting time/place: Monday and Wednesday, 5PM - 6:30PM Evans 35
Office hours: Monday, 7PM - 8PM, 835 Evans



Potentially Useful References:

C. Adams, The Knot Book (1994, W. H. Freeman)
D. Rolfsen, Knots and Links (1976, Publish or Perish)
W. B. R. Lickorish, An Introduction to Knot Theory (1997, Springer GTM)
J. Gross and T. Tucker, Topological Graph Theory (2003, Dover)


I am going to keep a brief record of what we do each day.

1.23.19 : We discussed some of the basic definitions in graph theory. We proved some simple facts about trees and gave a characterization of bipartite graphs.

1.28.19 : We started discussing graphs embedded on surfaces. We proved Euler's formula on the sphere and applied it to show that there are only 5 Platonic solids.

1.30.19 : We proved Kuratowski's theorem characterizing planar graphs. If you look closely, you will note that we also proved that any 3-connected graph can be drawn uniquely in the plane, that it can be drawn with convex faces, and that the edges can be drawn as straight lines. With a bit more work, the latter result can be generalized to arbitrary planar graphs, without the connectivity hypothesis. We also discussed the cycle double cover conjecture.

2.4.19 : We discussed the chromatic polynomial of a graph from a few different perspectives and showed some of its properties. We then proved the 5-color theorem for planar graphs and gasve some different formulations of the 4-color conjecture (now theorem). Here is a link to a chromatic polynomial reference. And here is a link to some info about June Huh who solved the problem regarding the unimodailty of the absolute values of the coefficients of the chromatic polynomial of a graph.

2.6.19 : We discussed the edge coloring statement that is equivalent to the 4-color theorem and proved equivalence. We computed some chromatic polynomials in a few different ways and we proved that the Euler characteristic of a graph does not in general depend on the graph or embedding, so long as the graph cuts the surface up into disks.

2.11.19 : We introduced rotation systems as a way of encoding graphs cellularly embedded on surfaces. We sketched a proof of the classification of surfaces (assuming the fact that surfaces are triangulizable). We mentioned the Heawood bound and then briefly introduced covering spaces.

2.13.19 : We showed how to construct all finite covers of a graph by equipping the graph with a permutation voltage assignment. We showed how a rotation system on a graph lifts to any finite cover on that graph. We used this to show how to construct all finite sheeted branched coverings of a compact surface by taking a graph cellularly embedded on the surface and equipping it with a permutation voltage assignment.

2.21.19 : We discussed the Riemann-Hurwitz formula and gave several examples of branched covers of surfaces. We showed how to construct branched covers of a surface given a cellularly embedded graph on the surface together with a permutation voltage assignment on the graph.

2.25.19 : We discussed knots, mirror images, reverses, sums of knots, and what it means for knots to be equivalent. We discussed diagrams of knots and stated Reidemeister's theorem. You might like this video.

2.27.19 : We discussed the genus of a knot, discussed Seifert's algorithm for constructing a Seifert surface for a knot from a knot projection and started in on the proof that the genus of a knot is additive w.r.t. connect sum. We mentioned Gabai's theorem that the genus of an alternatign knot can be obtained by applying Seifert's algrithm to an alternating projection of the knot.

3.4.19 : We finished the proof of the additivity of knot genus and concluded that genus 1 knots are prime. We introduced 3-colorability of a knot diagram and proved that it does not depend on the diagram that is used. We concluded that the trefoil knot it not the unknot.

3.12.19 : We introduced the Jones polynomial via the Kauffman bracket and did some calculations. We verified invariance under the Reidemeister moves. We gave an application of linking numbers to show that K_6 is intrisically linked. We introducde the unknotting number of a knot and stated some conjectures concerning it.

3.14.19 : We deduced some properties of the Jones polynomial and we defined the HOMFLYPT and Alexander polynomial via similar skein relations. We used the Jones polynomial to prove one of the Tait conjectures for alternating knots.

3.18.19 : We discussed the bridge number of a knot. Here is a modern proof of the additivity of bridge number that was mentioned. We discussed satellite knots - here are some interesting rather recent results that you might like to look at. Finally, we discussed the braid group from a few different angles (mapping class group, fundamental group of a configuration space, good old braids) and began talking about representing links as braid closures.

3.20.19 : We discussed the project logistics and I gave a couple of ideas for potential projects. We disscussed the Markov moves between different braids that preserve the braid closure and the proof that all links can be represented as closures of some braid. Here is a link to a fun movie illustrating the geometric knot theory concept of ropelength.

4.1.19 : We discussed the fundamental group of a space, presentations of groups, free products with amalgamtion, and the Seifert-van Kampen theorem. We tried to give lots of low dimensional examples and keep things concrete.

4.3.19 : We discussed Wirtinger's method for obtaining a presentation of the fundamental group of the complement of a knot given a diagram fo the knot. We used this to write down some presentations and we used the knot groups to tell some knots apart. We obtained presentations of the complements of torus knots using Seifert - van Kampen. We saw the the square knot and the granny knot have isomorphic fundamental groups.

4.8.19 : We discussed some examples of 3-manifolds and we flew around in them a bit. We started talking about obtaining 3-manifolds by doing surgery on a knot.

4.10.19 : We discussed lens spaces and Dehn surgery on links.

4.15.19 : We gave some examples of surgery of links and showed how to denote surgeries using surgery coefficients. We used this to give a method for giving surgery descriptions of finite cyclic covers of knot complements and illustrated this for the figure 8 knot.

4.17.19 : We considerde homotopical linking of links and gave some examples and nonexamples. We considered boundary linking of links and gave some examples and nonexamples. We showed that if a two component link is a boundary link then each knot is in the second derived subgroup of the fundamental group of the complement of the other knot.

4.19.19 : We discussed the construction of the infinite cyclic cover of a knot and uned it to show that the Whitehead link is not a boundary link. We started discussing 2-knots. We gave some nontrivial examples by way of Artin's spinning construction.

4.22.19 : We gave more examples of 2-knots and 2-links. We gave an example of a 2-component 2-link were each component is trivial but the link was not split. We gave an example of a 2-knot with torsion in the fundamental group of its complement.

4.24.19 : We discussed the projects.