New presentations of a link and virtual link
Liangxia Wan

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 are unknotted in time where is the number of crossings in each for .
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeometric and Algebraic Topology · Advanced Combinatorial Mathematics · Computational Geometry and Mesh Generation
