New presentations of a link and virtual link

TL;DR

This paper introduces new algebraic presentations for links and virtual links, along with algorithms to efficiently reduce crossings and determine unknots, including infinite families, in polynomial time.

Contribution

It presents novel algebraic systems for links and virtual links and proposes efficient reduction algorithms, enabling polynomial-time unknot recognition for certain families.

Findings

Algorithms reduce crossings in polynomial time

Unknots like Goeritz's and Thistlethwaite's are confirmed unknotted

Infinite family of knots unknotted in O(n^2) time

Abstract

New presentations of a link and a virtual link are introduced and algebraic systems on links and virtual links are constructed respectively. Based on the algebraic systems, Reduction Crossing Algorithms for them are proposed which are used to reduce the number of crossings in a link and virtual link. For known unknots, one can transform them into a trivial knot in a polynomial time by applying corresponding algorithm. As special consequences, Goeritz's unknot and Thistlethwaite's unknot are unknotted. Moreover, an infinite family of knots $K_{G_{2k,2l}}$ are unknotted in $O(n^2)$ time where $n$ is the number of crossings in each $K_{G_{2k,2l}}$ for $k,l\ge 0$.