Categorifying Quiver Linking/Unlinking using CoHA Modules

The knots-quivers correspondence is a relation between knot invariants and enumerative invariants of quivers, which in particular translates the knot operations of linking and unlinking to a certain mutation operation on quivers. In this paper we show that the moduli spaces of a quiver and its linking/unlinking are naturally related, giving a purely representation theoretic interpretation of these operations. We obtain a relation between the cohomologies of these spaces which is moreover compatible with a natural action of the Cohomological Hall Algebra. The result is a categorification of quiver linking/unlinking at the level of CoHA modules.