Liam Watson’s research while affiliated with University of British Columbia and other places

When restricted to alternating links, both Heegaard Floer and Khovanov homology concentrate along a single diagonal $\delta$ -grading. This leads to the broader class of thin links that one would like to characterize without reference to the invariant in question. We provide a relative version of thinness for tangles and use this to characterize thinness via tangle decompositions along Conway spheres. These results bear a strong resemblance to the L-space gluing theorem for three-manifolds with torus boundary. Our results are based on certain immersed curve invariants for Conway tangles, namely the Heegaard Floer invariant $\operatorname {HFT}$ and the Khovanov invariant $\widetilde {\operatorname {Kh}}$ that were developed by the authors in previous works.

We prove an equivariant version of the Cosmetic Surgery Conjecture for strongly invertible knots. Our proof combines a recent result of Hanselman with the Khovanov multicurve invariants Kh~${\widetilde{{{\,\textrm{Kh}\,}}}}$ and BN~${\widetilde{{{\,\textrm{BN}\,}}}}$. We apply the same techniques to reprove a result of Wang about the Cosmetic Crossing Conjecture and split links. Along the way, we show that Kh~${\widetilde{{{\,\textrm{Kh}\,}}}}$ and BN~${\widetilde{{{\,\textrm{BN}\,}}}}$ detect if a Conway tangle is split.

In previous work we introduced a Khovanov multicurve invariant $\operatorname{\widetilde{Kh}}$ associated with Conway tangles. Applying ideas from homological mirror symmetry we show that $\operatorname{\widetilde{Kh}}$ is subject to strong geography restrictions: Every component of the invariant is linear, in the sense that it admits a lift to a curve homotopic to a straight line in an appropriate planar cover of the tangle boundary.

We prove an equivariant version of the Cosmetic Surgery Conjecture for strongly invertible knots. Our proof combines a recent result of Hanselman with the Khovanov multicurve invariants $\widetilde{\operatorname{Kh}}$ and $\widetilde{\operatorname{BN}}$. We apply the same techniques to reprove a result of Wang about the Cosmetic Crossing Conjecture and split links. Along the way, we show that $\widetilde{\operatorname{Kh}}$ and $\widetilde{\operatorname{BN}}$ detect if a Conway tangle is split.

When restricted to alternating links, both Heegaard Floer and Khovanov homology concentrate along a single diagonal $\delta$-grading. This leads to the broader class of thin links that one would like to characterize without reference to the invariant in question. We provide a relative version of thinness for tangles and use this to characterize thinness via tangle decompositions along Conway spheres. These results bear a strong resemblance to the L-space gluing theorem for three-manifolds with torus boundary. Our results are based on certain immersed curve invariants for four-ended tangles, namely the Heegaard Floer invariant $\operatorname{HFT}$ and the Khovanov invariant $\operatorname{\widetilde{Kh}}$ that were developed by the authors in previous works. Applying ideas from homological mirror symmetry, we show that $\operatorname{\widetilde{Kh}}$ is subject to the same strong geography restrictions that were already known for $\operatorname{HFT}$.

There is a one-to-one correspondence between strong inversions on knots in the three-sphere and a special class of four-ended tangles. We compute the reduced Khovanov homology of such tangles for all strong inversions on knots with up to 9 crossings, and discuss these computations in the context of earlier work by the second author. In particular, we provide a counterexample to [Conjecture 29, arXiv:1311.1085] as well as a refinement of and additional evidence for [Conjecture 28, arXiv:1311.1085].

When $\mathbf{k}$ is a field, type D structures over the algebra $\mathbf{k}[u,v]/(uv)$ are equivalent to immersed curves decorated with local systems in the twice-punctured disk [arXiv:1910.14584]. Consequently, knot Floer homology, as a type D structure over $\mathbf{k}[u,v]/(uv)$, can be viewed as a set of immersed curves. With this observation as a starting point, given a knot K in $S^3$, we realize the immersed curve invariant $\widehat{\mathit{HF}}(S^3 \setminus \mathring{\nu}(K))$ [arXiv:1604.03466] by converting the twice-punctured disk to a once-punctured torus via a handle attachment. This recovers a result of Lipshitz, Ozsv\'ath, and Thurston [arXiv:0810.0687] calculating the bordered invariant of $S^3 \setminus \mathring{\nu}(K)$ in terms of the knot Floer homology of K.