Writhe-like invariants of alternating links

TL;DR

This paper introduces new simple-to-calculate invariants derived from Seifert graphs that distinguish alternating links and remain invariant under flypes, providing an easier alternative to complex invariants like the Jones polynomial.

Contribution

The paper presents novel writhe-like invariants based on Seifert graphs that are invariant among reduced alternating link diagrams and are computationally elementary.

Findings

New invariants distinguish many alternating links effectively.

Invariants are easy to compute and compare with existing invariants.

Derived a criterion for strongly invertible rational links.

Abstract

It is known that the writhe calculated from any reduced alternating link diagram of the same (alternating) link has the same value. That is, it is a link invariant if we restrict ourselves to reduced alternating link diagrams. This is due to the fact that reduced alternating link diagrams of the same link are obtainable from each other via flypes and flypes do not change writhe. In this paper, we introduce several quantities that are derived from Seifert graphs of reduced alternating link diagrams. We prove that they are "writhe-like" invariants in the sense that they are also link invariants among reduced alternating link diagrams. The determination of these invariants are elementary and non-recursive so they are easy to calculate. We demonstrate that many different alternating links can be easily distinguished by these new invariants, even for large, complicated knots for which other invariants such as the Jones polynomial are hard to compute. As an application, we also derive an if and only if condition for a strongly invertible rational link.