[Paper Review] The cosmetic crossing conjecture for split links

Joshua Wang|arXiv (Cornell University)|Jun 1, 2020

Geometric and Algebraic Topology45 references4 citations

TL;DR

This paper proves the generalized cosmetic crossing conjecture for split links by showing that band sums of a split two-component link along a nontrivial band yield infinitely many distinct unoriented knots when full twists are added to the band. Using Khovanov homology over 𝔽₂, the authors demonstrate that the bigraded homology groups of these knots are all distinct, while Heegaard and instanton knot Floer homologies remain invariant under twisting, thus confirming the conjecture's prediction for split links.

ABSTRACT

Given a band sum of a split two-component link along a nontrivial band, we obtain a family of knots indexed by the integers by adding any number of full twists to the band. We show that the knots in this family have the same Heegaard knot Floer homology and the same instanton knot Floer homology. In contrast, a generalization of the cosmetic crossing conjecture predicts that the knots in this family are all distinct. We verify this prediction by showing that any two knots in this family have distinct Khovanov homology. Along the way, we prove that each of the three knot homologies detects the trivial band.

Motivation & Objective

• To verify the generalized cosmetic crossing conjecture for split two-component links.
• To show that band sums of split links along nontrivial bands produce infinitely many distinct unoriented knots under full twists.
• To demonstrate that Khovanov homology detects the nontriviality of the band and distinguishes the twisted knots.
• To establish that Heegaard and instanton knot Floer homologies remain invariant under full twists, contrasting with Khovanov homology's sensitivity.

Proposed method

• Construct a family of knots {K_{b+n/2}} by adding n half-twists to a nontrivial band in a split two-component link L.
• Use ribbon concordances and planar movie presentations to relate Khovanov homology of K_{b+n/2} to that of the connected sum K_{#} and the link L.
• Apply unoriented skein exact triangles in Khovanov homology over 𝔽₂ to analyze the behavior of homology under twisting.
• Leverage functoriality and grading shifts to show that Khovanov homology groups shift by h^n q^{2n} H_b, where H_b is a nonzero bigraded vector space.
• Prove that the isomorphism type of Khovanov homology over 𝔽₂ is an invariant of unoriented knot type, enabling distinction of knots.
• Use the fact that Kh(K_b) ≅ Kh(K_#) ⊕ H_b with H_b ≠ 0 and H_{b+n/2} ≅ h^n q^{2n} H_b to show distinctness across all n ∈ ℤ.

Experimental results

Research questions

• RQ1 Are all knots obtained by adding full twists to a nontrivial band in a split two-component link distinct as unoriented knots?
• RQ2 Does Khovanov homology detect the nontriviality of the band in a band sum of a split link?
• RQ3 Why do Heegaard and instanton knot Floer homologies remain invariant under full twists, while Khovanov homology does not?
• RQ4 Can the generalized cosmetic crossing conjecture be verified for split links using Khovanov homology?
• RQ5 Is the bigraded structure of Khovanov homology over 𝔽₂ sufficient to distinguish all knots in the twisted family?

Key findings

• The knots K_{b+n/2} for n ∈ ℤ are all distinct as unoriented knots, confirming the generalized cosmetic crossing conjecture for split links.
• Khovanov homology over 𝔽₂ detects the nontriviality of the band, as H_b is nonzero and of finite dimension.
• The bigraded vector space Kh(K_{b+n/2}) is isomorphic to Kh(K_#) ⊕ h^n q^{2n} H_b, showing a consistent shift under twisting.
• The isomorphism type of Kh(K_{b+n/2}) over 𝔽₂ is distinct for each n ∈ ℤ, proving that the knots are not isotopic.
• Heegaard and instanton knot Floer homologies remain unchanged under full twists, indicating invariance under band twisting for split links.
• The Khovanov homology of the connected sum K_# is isomorphic to the direct sum of Kh(K_#) and H_b, with H_b capturing the twisting effect.