Representing knots with graph theory

I study knot theory. I find that the work and how it's taught is very different and not very accessible even at the introductory level. I had a hard time understanding what makes a "positive (right)" crossing distinct from an overcrossing and distinct from a negative crossing.

It is very common that the articles give a very general definition without a single example, when it would have been much easier to understand the general theory from a number of examples only and letting the reader draw a general conclusion from a number of examples instead of the other way around. I think that is too common in mathematics that the mathematician wants to present the generalization without a single example which makes it very difficult to understand the examples. I would have liked to learn by example.

Now I understand that many but not every knot can be represented by a Gauss diagram and the Gauss diagrams become incomprehensible for more complicated knots. It makes it no simpler to compare knots at all. How can I represent a knot with mathematics instead of illustrations? Assume I have knots of the classical type, with crossings and if the knots are oriented, can I represent these kinds of knots with graph theory or why not? A crossing can be represented as a node from graph theory and we just need to add if the crossing is positive or negative. Why would it not work? Then the knots can be represented with methods from regular graph theory and proving something about knots will be the same as proving something about the graphs.

1 Answer

Every knot can be represented by a Gauss code, which is what Gauss diagrams illustrate (I'm not sure what you're referring to how not every knot can be represented by a Gauss diagram -- what's true is that not every Gauss diagram represents a knot). The situation is very similar to how graphs tend to be drawn pictorially, yet it's understood that they are given by the data $$(V,E)$$.

It's common representing knots as graphs, though you have to be careful to represent the cyclic ordering of edges around vertices so you know how to embed the graph in a plane again. In fact, representing knots as graphs is essentially how knots are enumerated to create knot tables.

I've written about how to represent knots at https://math.berkeley.edu/~kmill/2019_8_25/gauss_dt_codes.html

A small technicality: knots are circles embedded in 3-space, and the idea is that (1) every combinatorial description of a knot can be realized as a knot and (2) every knot is isotopic to one of these realizations. We usually only care about isotopy classes of knots so this is ok, but I thought I'd mention that combinatorial descriptions forget about the way the knot actually sat in 3-space.

There's also another way to represent knots as graphs, and that's the Tait graph. The regions of a knot diagram can be checkerboard colored, and for one of the two colors you can associate a planar graph with signed edges. Or, in reverse, given a planar graph with signed edges, you can associate crossings to the edges based on the sign and then connect everything up around the vertices.

• I misunderstood the text in Sam Nelson's article ("Not every Gauss diagram corresponds to a classical knot."). But if two knots are the same, how can they have different Gauss codes? What is the problem then to see if two knots are the same, if they can be represented with two Gauss codes or two graphs? Why would the Gauss codes be different and why would the problem be any different than proving an isomorphism, which is sometimes difficult but a known problem? Jun 27, 2021 at 6:57
• @NiklasR. Two knot diagrams are equivalent if they are isomorphic after applying zero or more Reidemeister moves -- this is the difficulty. There are various (slow) algorithms for telling whether two knots diagrams are equivalent. For example, this paper tells you the maximum length of a sequence of Reidemeister moves you could exhaustively try: people.maths.ox.ac.uk/lackenby/rei17611.pdf Jun 27, 2021 at 16:00