Khovanov multicurves are linear

美容手术与 Khovanov 多重曲线

Cosmetic operations and Khovanov multicurves

本文结合 Hanselman 的结果与 Khovanov 多重曲线不变量，证明强可逆纽结的等变美容手术猜想，并重新证明美容交叉猜想相关结果及 Conway 缠结分裂性的可检测性。

双缠结的Heegaard Floer多重曲线

Heegaard Floer multicurves of double tangles

本文给出了从纽结补集Heegaard Floer多重曲线计算双缠结不变量的公式，并借此发现纽结Floer同调与Khovanov同调在卫星运算下行为迥异，同时给出L空间纽结的新刻画及首个Floer同调为薄的卫星纽结。

Immersed curves in Khovanov homology

We give a geometric interpretation of Bar-Natan's universal invariant for the class of tangles in the 3-ball with four ends: we associate with such 4-ended tangles $T$ multicurves $\widetilde{\operatorname{BN}}(T)$, that is, collections of immersed curves with local systems in the 4-punctured sphere. These multicurves are tangle invariants up to homotopy of the underlying curves and equivalence of the local systems. They satisfy a gluing theorem which recovers the reduced Bar-Natan homology of links in terms of wrapped Lagrangian Floer theory. Furthermore, we use $\widetilde{\operatorname{BN}}…

关于Khovanov同调中突变不变性的研究

On mutation invariance in Khovanov homology

本文研究了Khovanov同调在Conway突变下的不变性，并通过同调镜像对称方法对相关不变量进行了分类。

A family of concordance homomorphisms from Khovanov homology

By considering a version of Khovanov homology incorporating both the Lee and $E(-1)$ differentials, we construct a $1$-parameter family of concordance homomorphisms similar to the Upsilon invariant from knot Floer homology. This invariant gives lower bounds on the slice genus and can be used to prove that certain infinite families of pretzel knots are linearly independent in the smooth concordance group.

Simplified Khovanov-Rozansky Chain Complexes of Open 2-braids

Motivated by the works of Krasner [arXiv:0801.4018] and Lobb [arXiv:1103.1412], we simplify the Khovanov-Rozansky chain complexes of open 2-braids. As an application, we show that, for a knot containing a "long" 2-braid, the sl(N) Rasmussen invariant of this knot depends linearly on the length of this 2-braid. We refine this result for (2,2k+1) cable knots and, as a simple corollary, compute the sl(2) Rasmussen invariants of (2,2k+1) cables of slice and amphicheiral knots.