Clock Moves and Alexander Polynomial of Plane Graphs

TL;DR

This paper introduces clock moves for spanning trees in plane graphs, developing a model for the Alexander polynomial that confirms the trapezoidal conjecture for planar singular knots and offers new insights into Fox's conjecture.

Contribution

It presents a novel clock move concept for spanning trees and constructs an Alexander polynomial model for plane graphs, advancing knot theory understanding.

Findings

Confirmed the trapezoidal conjecture for planar singular knots

Developed a spanning tree model of the Alexander polynomial

Provided new insights into Fox's original conjecture on alternating knots

Abstract

In this paper, we introduce a notion of clock moves for spanning trees in plane graphs. This enables us to develop a spanning tree model of an Alexander polynomial for a plane graph and prove the unimodal property of its associate coefficient sequence. In particular, this confirms the trapezoidal conjecture for planar singular knots and gives new insights to Fox's original conjecture on alternating knots.