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David Avellan: Some algebraic methods in knot theory

Bachelor thesis presentation

Time: Tue 2022-08-23 10.30 - 11.30

Location: Albano, House 1, Room N

Doctoral student: David Avellan

Abstract:

This thesis introduces some knot theory, focusing on the knot group and related ideas. Knots are defined as smooth embeddings of the circle in three-dimensional space, and the initial part of the thesis is dedicated to different notions of knot equivalence as well as certain geometric concepts. The knot group of a knot is defined as the fundamental group of the complement of the knot, and much of the thesis revolves around proving that knots are inequivalent by showing that they have non-isomorphic knot groups. A proof is given for a method of calculating a presentation of a knot group, the socalled ‘Wirtinger presentation’. Next, an infinite class of knots called ‘torus knots’ is introduced, and their knot groups are analyzed and classified. The knot group fails to distinguish some knots that are nonetheless different, a problem adressed in the final part of the thesis through the so-called ‘peripheral system’, which augments the knot group with additional information about the knot, giving a more powerful invariant. Two applications of the peripheral system are presented: showing that no torus knot is equivalent to its mirror image, and proving two other specific knots inequivalent.