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Cosmetic operations and Khovanov multicurves.

Artem Kotelskiy1, Tye Lidman2, Allison H Moore3

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We proved an equivariant version of the Cosmetic Surgery Conjecture for strongly invertible knots using Khovanov multicurve invariants. These invariants also help detect split Conway tangles and reprove results on the Cosmetic Crossing Conjecture.

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Area of Science:

  • Knot theory
  • Low-dimensional topology
  • Algebraic topology

Background:

  • The Cosmetic Surgery Conjecture is a major open problem in knot theory.
  • Strongly invertible knots and Conway tangles are important classes of knots and links.
  • Khovanov multicurve invariants provide powerful tools for distinguishing knots and links.

Purpose of the Study:

  • To prove an equivariant version of the Cosmetic Surgery Conjecture for strongly invertible knots.
  • To apply new techniques to reprove Wang's result on the Cosmetic Crossing Conjecture and split links.
  • To demonstrate the utility of Khovanov multicurve invariants in detecting split Conway tangles.

Main Methods:

  • Combining Hanselman's recent results with Khovanov multicurve invariants ( and ).
  • Utilizing equivariant versions of topological invariants.
  • Applying these methods to analyze Conway tangles and split links.

Main Results:

  • An equivariant version of the Cosmetic Surgery Conjecture is proven for strongly invertible knots.
  • The Khovanov multicurve invariants ( and ) are shown to detect if a Conway tangle is split.
  • Wang's result concerning the Cosmetic Crossing Conjecture and split links is reproved.

Conclusions:

  • The Khovanov multicurve invariants are effective tools for addressing fundamental questions in knot theory.
  • The techniques developed offer a new approach to studying knot concordance and related problems.
  • This work advances the understanding of topological invariants and their applications in low-dimensional topology.