On the warping sum of knots

TL;DR

This paper characterizes knots with a warping sum of 3 or less and provides examples of knots with a warping sum of 4, advancing understanding of knot complexity measures.

Contribution

It introduces a characterization of knots with low warping sum and identifies specific knots with warping sum equal to 4, a novel contribution to knot theory.

Findings

Knots with warping sum ≤ 3 are fully characterized.

Examples of knots with warping sum = 4 are provided.

The study enhances understanding of knot complexity related to warping sum.

Abstract

The warping sum $e(K)$ of a knot $K$ is the minimal value of the sum of the warping degrees of a minimal diagram of $K$ with both orientations. In this paper, knots $K$ with $e(K) \le 3$ are characterized, and some knots $K$ with $e(K)=4$ are given.