Visual Algebraic proofs for Unknot Detection

A knot diagram looks like a two-dimensional drawing of aknotted rubberband. Proving that a given knot diagram can be untangled(that is, is a trivial knot, called an unknot) is one of the most famousproblems of knot theory. For a small knot diagram, one can try to finda sequence of untangling moves explicitly, but for a larger knot diagramproducing such a proof is difficult, and the produced proofs are hardto inspect and understand. Advanced approaches use algebra, with anadvantage that since the proofs are algebraic, a computer can be usedto produce the proofs, and, therefore, a proof can be produced evenfor large knot diagrams. However, such produced proofs are not easy toread and, for larger diagrams, not likely to be human readable at all.We propose a new approach combining advantages of these: the proofsare algebraic and can be produced by a computer, whilst each part ofthe proof can be represented as a reasonably small knot-like diagram(a new representation as a labeled tangle diagram), which can be easilyinspected by a human for the purposes of checking the proof and findingout interesting facts about the knot diagram

Towards human readability of automated unknottedness proofs

© 2018 CEUR-WS. All rights reserved. When is a knot actually unknotted? How does one convince a human reader of the correctness of an answer to this question for a given knot diagram? For knots with a small number of crossings, humans can be efficient in spotting a sequence of untangling moves. However, for knot diagrams with hundreds of crossings, computer assistance is necessary. There have been recent developments in algorithms for both (indirectly) (i) detecting unknotedness and (directly) (ii) producing such sequences of untangling moves. Automated reasoning can be applied to (i) and, to some extent, (ii), but the computer output is not necessarily human-readable. We report on work in progress towards bridging the gap between the computer output and human readability, via generating human-readable visual proofs of unknottedness