Twisting braids and surfaces

This thesis covers two twisting notions, namely that of Dehn twists and that of Fox’s twist moves. More precisely, in the first part we characterise closed embedded discs in surfaces that are bounded by a circuit curve system in terms of a particular relation called the cycle relation between the associated Dehn twists. We also characterise bouquets of curves in a similar vein. In the process, we classify all circuits of curves in terms of whether the corresponding Dehn twists generate an Artin group. In the second part we relate Przytycky’s obstructions on the HOMFLY polynomial for two links to be related by twist moves to the crossing number and the braid index. We also estimate the cobordism distance between connected sum powers of trefoil or hexafoil links and three strand torus links.

2024

dissertation

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