Clock Moves and Alexander Polynomial of Plane Graphs

Wenbo Liao, Zhongtao Wu

Published 2024-10-21, updated 2024-10-22Version 2

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.