0
votes
1answer
57 views

Link diagrams and Reidemeister moves

I am studying Knots on "Algebraic Graph Theory" written by Godsil & Royle. They state the following theorem: $\underline{Theorem}$ Two link diagrams determine the same link if and only if one can ...
2
votes
0answers
71 views

How do you specify a link to a blind combinatorialist?

Regular projections of links look like graphs in the plane. So I'm wondering if it would be possible to specify a link up to isotopy with purely combinatorial data about this graph. If so, what kind ...
1
vote
0answers
15 views

Determine the multiplicity of knots for a graph

Here are my two questions: Given a finite connected non-oriented planar graph, is there a way to determine whether or not it is possible to derive a single non-trivial knot diagram from this graph, ...
1
vote
1answer
56 views

Uniqueness of Seifert graphs

If we make the bands and disks of a Seifert surface really small and really thin the surface collapses to a graph. It is called a Seifert graph. If it is not a directed and weighted graph, can we ...
1
vote
0answers
49 views

Hamiltonian of one and two unknots

Recently I calculated the Ising Hamiltonian of a Hopf link. First, I colored the Hopf link in a checker board pattern and drew the Seifert surface from it. Considering the shaded regions as vertices ...
6
votes
1answer
150 views

Knots and graphs

Every knot gives rise to a number of 4-regular planar graphs - by regular projections onto the plane - which just have to be enriched by an over/under flag for every vertex to be able to reconstruct ...
2
votes
1answer
70 views

Computational complexity of unknotting problem?

The Wikipedia article on the unknotting problem says "a major unresolved challenge is to determine [...] whether the problem lies in the complexity class P". It mentions some work towards this result ...
4
votes
1answer
160 views

Assigning alternate crossings to closed curves

This is a minor curiosity that I've been wondering about. Suppose that we draw a closed curve in the plane and that this curve intersects itself several times, but never twice in one spot. We can knot ...