Automated Reasoning for Knot Semigroups and \pi π -orbifold Groups of Knots

The paper continues the first author’s research which shows that automatic reasoning is an effective tool for establishing properties of algebraic constructions associated with knot diagrams. Previous research considered involutory quandles (also known as keis) and quandles. This paper applies automated reasoning to knot semigroups, recently introduced and studied by the second author, and \(\pi \)-orbifold groups of knots. We test two conjectures concerning knot semigroups (specifically, conjectures aiming to describe knot semigroups of diagrams of the trivial knot and knot semigroups of 4-plat knot diagrams) on a large number of examples. These experiments enable us to formulate one new conjecture. We discuss applications of our results to a classical problem of the knot theory, determining whether a knot diagram represents the trivial knot.

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