A Transfer Matrix approach to the Enumeration of Knots

TL;DR

This paper introduces a transfer matrix method for enumerating alternating knots, enabling the counting of complex knot structures up to high order and exploring their asymptotic behavior.

Contribution

It presents a novel transfer matrix approach for knot enumeration, extending the ability to count prime alternating tangles to higher orders than previous methods.

Findings

Successfully enumerated prime alternating tangles up to order 22

Observed large-order behavior consistent with existing conjectures

Provided a new computational framework for knot enumeration

Abstract

We propose a new method to enumerate alternating knots using a transfer matrix approach. We apply it to count numerically various objects, including prime alternating tangles with two connected components, up to order 18--22, and comment on the large-order behavior in connection with one of the authors' conjecture.