1. Sriram Raghunath, A note on the Lipshitz-Sarkar spectral sequence for the Khovanov homology of strongly invertible knots. (In preparation, draft available on request)
Abstract: We prove that the Lipshitz-Sarkar spectral sequence for the Khovanov homology of a directed strongly invertible knot is independent of the choice of intravergent diagram used to represent the knot, by showing the invariance of the spectral sequence under equivariant Reidemeister moves on the intravergent knot diagram. We also show that the final page of their spectral sequence splits as the direct sum of the homology of isomorphic chain complexes, giving a 'reduced' perspective of the final page. We compute some salient examples using a modification of Champ Davis' annular Khovanov homology calculator.
2. Kristen Hendricks, Cheuk Yeu Mak, and Sriram Raghunath, Symplectic annular Khovanov homology and fixed point localizations (Submitted) (arXiv link)
Abstract: We introduce a new version of symplectic annular Khovanov homology and establish spectral sequences from (i) the symplectic annular Khovanov homology of a knot to the link Floer homology of the lift of the annular axis in the double branched cover; (ii) the symplectic Khovanov homology of a two-periodic knot to the symplectic annular Khovanov homology of its quotient; and (iii) the symplectic Khovanov homology of a strongly invertible knot to the cone of the axis-moving map between the symplectic annular Khovanov homology of the two resolutions of its quotient.
3. Tejas Kalelkar and Sriram Raghunath, Bounds on Pachner moves and systoles of cusped 3–manifolds. Algebraic & Geometric Topology 22:6 (2022), pp. 2951–2996. doi:10.2140/agt.2022.22.2951 (Published version) (arXiv link)
Abstract: Any two geometric ideal triangulations of a cusped complete hyperbolic 3-manifold M are related by a sequence of Pachner moves through topological triangulations. We give a bound on the length of this sequence in terms of the total number of tetrahedra and a lower bound on dihedral angles. This leads to a naive but effective algorithm to check if two hyperbolic knots are equivalent, given geometric ideal triangulations of their complements. Given a geometric ideal triangulation of M, we also give a lower bound on the systole length of M in terms of the number of tetrahedra and a lower bound on dihedral angles.