Sigma-Adequate Link Diagrams and the Tutte Polynomial

TL;DR

This paper characterizes sigma-adequacy in link diagrams using Tutte polynomials and Tait graphs, providing methods to identify all sigma-adequate states and conditions for sigma-adequacy and sigma-homogeneity.

Contribution

It introduces a new characterization of sigma-adequacy via Tutte polynomials and offers a method to find all sigma-adequate states of a link diagram.

Findings

Number of sigma-adequate states is bounded by spanning trees in the Tait graph.

A sum of products of Tutte polynomials represents the symmetrized Tutte polynomial.

Necessary and sufficient conditions for sigma-adequacy and sigma-homogeneity are established.

Abstract

In this paper, we characterize the sigma-adequacy of a link diagram in two ways: in terms of a certain edge subset of its Tait graph and in terms of a certain product of Tutte polynomials. Furthermore, we show that the symmetrized Tutte polynomial of the Tait graph of a link diagram can be written as a sum of these products of Tutte polynomials, where the sum is over the sigma-adequate states of the given link diagram. Using this state sum, we show that the number of sigma-adequate states of a link diagram is bounded above by the number of spanning trees in its associated Tait graph. By combining results, we give a method to find all of the sigma-adequate states of a link diagram. Finally, we give necessary and sufficient conditions for a link diagram to be sigma-adequate and sigma-homogeneous (also called homogeneously adequate) with respect to a given state.