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

Describing semigroups with defining relations of the form xy=yz xy and yx=zy and connections with knot theory

We introduce a knot semigroup as a cancellative semigroup whose defining relations are produced from crossings on a knot diagram in a way similar to the Wirtinger presentation of the knot group; to be more precise, a knot semigroup as we define it is closely related to such tools of knot theory as the twofold branched cyclic cover space of a knot and the involutory quandle of a knot. We describe knot semigroups of several standard classes of knot diagrams, including torus knots and torus links T(2, n) and twist knots. The description includes a solution of the word problem. To produce this description, we introduce alternating sum semigroups as certain naturally defined factor semigroups of free semigroups over cyclic groups. We formulate several conjectures for future research