2025/10/10 by Hayato Imori, Imori, Hayato, Taketo Sano +5
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.2510.09399
This paper is a continuation of our previous work, where we defined an embedded cobordism map on the instanton cube complex that recovers the cobordism maps both in Khovanov homology and singular instanton theory. In this paper, we extend this construction to immersed cobordisms, where we define an immersed cobordism map on Khovanov homology and prove that it is compatible with the immersed cobordism map on singular instanton homology. We give two applications: (i) For any smooth, oriented concordance C from a two-bridge torus knot, the induced map Kh(C) on Khovanov homology is injective, and its left inverse is given by the reversal of C. (ii) Any pair of relatively exotic surfaces in D4 that are detected by the embedded cobordism map in Kh remain exotic even after applying any number of positive twist moves.