The warping polynomial of a knot diagram

TL;DR

This paper introduces the warping polynomial for oriented knot diagrams, characterizes it, and explores its relation to knot properties like span, non-triviality, and alternating nature.

Contribution

It defines the warping polynomial, characterizes it, and establishes a relationship between the span of a knot and its topological features.

Findings

Span of a knot is one if and only if it is non-trivial and alternating.

Provides an inequality relating the span and the dealternating number.

Characterizes the warping polynomial for oriented knot diagrams.

Abstract

We introduce the warping polynomial of an oriented knot diagram. In this paper, we characterize the warping polynomial, and define the span of a knot to be the minimal span of the warping polynomial for all diagrams of the knot. We show that the span of a knot is one if and only if it is non-trivial and alternating, and we give an inequality between the span and the dealternating number.