Cobordism maps in Khovanov homology and singular instanton homology II

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 $\mathit{Kh}(C)$ on Khovanov homology is injective, and its left inverse is given by the reversal of $C$. (ii) Any pair of relatively exotic surfaces in $D^4$ that are detected by the embedded cobordism map in $\mathit{Kh}$ remain exotic even after applying any number of positive twist moves.