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Clock Moves and Alexander Polynomial of Plane Graphs

Wenbo Liao, Zhongtao Wu

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.

Clock Moves and Alexander Polynomial of Plane Graphs

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.

Paper Structure

This paper contains 6 sections, 19 theorems, 26 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Background and Parallel edge replacements
  3. Clock moves and clock theorem
  4. A spanning tree model
  5. Trapezoidal conjecture for plane graphs
  6. Fox's trapezoidal conjecture for alternating knots

Key Result

Theorem 1.1

Suppose $G$ is a plane graph and $\Delta_G(t)\doteq \sum_{i=1}^n a_it^i$ with $a_1=a_n\neq 0$. Then all the coefficients $a_i$'s are positive. Here and henceforth, the notation $\doteq$ denotes an equality up to a power of $t$.

Figures (22)

  • Figure 1: An orientation satisfying (left) / not satisfying (right) the transverse condition.
  • Figure 3: A parallel replacement move on an edge of color $n$
  • Figure 4: The corresponding Kauffman states resulting from a parallel replacement move on an edge of color $n$.
  • Figure 5: The counter-clockwise clock move turning $e$ to $e'$. Blue edges are contained in the spanning trees while dotted edges are in the complement of the spanning trees.
  • Figure 6: Stating from $v$ and walking backwards, we obtain a new cycle in $H'$ consisting of a path $p$, the "left" half of the original cycle $C$, and the new edge $e'$.
  • ...and 17 more figures

Theorems & Definitions (53)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • proof
  • ...and 43 more