Surjectivity conjecture for saddle cobordisms between braid diagrams

Let Dn,miD_{n,m}^i and En,miE_{n,m}^i be the diagrams defined in the paper, and consider the saddle cobordism

Dn,n1iDn,n1i1D_{n,n-1}^i\longrightarrow D_{n,n-1}^{i-1}

at the crossing corresponding to the last letter σi\sigma_i. Surjectivity conjecture. This saddle cobordism induces a surjection in rational Khovanov homology. Equivalently, the associated exact triangle splits into a short exact sequence. An affirmative answer would express the rational Khovanov homology of T(n,n)T(n,n) in terms of that of torus knots T(n+1,n)T(n'+1,n'); the supplied text gives no resolution status.

Sources & referencesView supporting material

Primary source

Qiuyu Ren, “Lee filtration structure of torus links”, arXiv:2305.16089 (2024).

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