# On the equivalence of virtual knots with at most six crossings.

As a summer project for Professor Dror Bar-Natan, Jeremy Green created a table of virtual knots (link). Quoting from the about page:

The program generates a list of Gauss codes, then, via Reidemeister moves, determines which are equivalent to each other. Using an Athlon XP 2500+ with 512 MB RAM, the enumeration can be done up to 8 crossing oriented virtual knots, using 10 GB of disk, in about 24 hours. This 8 crossing enumeration is only useful for knots up to 6 crossings because of the need for higher crossing intermediates when finding relationships.

This last sentence seems to be saying that the 8 crossing enumeration is useful for knots up to 6 crossings. That is, Green seems to be claiming that if $J$ and $K$ are two virtual knot diagrams with at most $6$ crossings, then the following are equivalent:

a) There exists a (finite) sequence $K_1,K_2,\cdots,K_n$ of virtual knot diagrams, each obtainable from the next by one of the (regular or virtual) Reidemeister moves, such that $J=K_1$ and $K=K_n$.

b) There exists a (finite) sequence $J_1,J_2,\cdots,J_m$ of virtual knot diagrams, each obtainable from the next by one of the (regular or virtual) Reidemeister moves, such that $J=J_1$ and $K=J_m$, and such that $J_i$ has at most $8$ crossings for each $i\in\{1,2,\cdots,m\}$.

I am looking for a reference for this claim.

• Perhaps the only reference is the about page and Green’s computer code. – Adam Lowrance Nov 28 '17 at 22:26
• For instance, the following papers cite Green's work, but only refer to the website that you linked: arXiv:0907.2215, arXiv:1110.4911, and arXiv:1506.01726. – Adam Lowrance Nov 28 '17 at 22:33