Quantization of the crossing number of a knot diagram

TL;DR

This paper introduces the warping crossing polynomial for oriented knot diagrams, explores its properties across different states of plane curves, and establishes the existence of canonical orientations based on crossing parity.

Contribution

It presents a novel polynomial invariant derived from warping degrees and analyzes canonical orientations for plane curves with even or odd crossings.

Findings

Every closed transversely intersected plane curve with even crossings has two canonical orientations.

Every based closed transversely intersected plane curve with odd crossings has two canonical orientations.

The warping crossing polynomial provides a new perspective on knot diagram invariants.

Abstract

We introduce the warping crossing polynomial of an oriented knot diagram by using the warping degrees of crossing points of the diagram. Given a closed transversely intersected plane curve, we consider oriented knot diagrams obtained from the plane curve as states to take the sum of the warping crossing polynomials for all the states for the plane curve. As an application, we show that every closed transversely intersected plane curve with even crossing points has two independent canonical orientations and every based closed transversely intersected plane curve with odd crossing points has two independent canonical orientations.