Jones-cosmetic tangle replacement

Pamela Shah (University of British Columbia)

Monday 24th April, 2023 16:00-17:00 Maths 311B

Abstract

Bar-Natan lists pairs of knots that have identical Jones polynomial and distinct Khovanov homology. A subset of this list consists of rational knots paired with certain rational tangle closures of the (3,-2) pretzel tangle. Each pair of this form is related by replacing the (3,-2) pretzel tangle with a rational tangle: an operation that leaves the Jones polynomial unchanged under favourable conditions. We investigate a pair on this list, 10₁₃₂ and the cinquefoil 5₁, using immersed curve invariants developed by Kotelskiy, Watson and Zibrowius. Using this viewpoint we see how the Jones polynomials of 10₁₃₂ and the cinquefoil 5₁ agree and how the Khovanov invariants differ. This leads us to the question: For which rational knots does there exist a rational closure of the (3,-2) pretzel tangle with the same Jones polynomial? In particular, it is instructive to warm up with: does there exist a rational closure of (3,-2) pretzel tangle with the same Jones polynomial as the unknot? We explain why this cannot be the case, and gesture towards some further questions and work in progress.

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