Quandle presentations of surface knots in 4-manifolds and bridge numbers

Abstract

The fundamental quandle is an invariant for distinguishing surface knots, yet computable presentations have traditionally been limited to surfaces embedded in the $4$-sphere. Building on the framework of banded unlink diagrams introduced by Hughes, Kim, and Miller, we give a Wirtinger type presentation of the fundamental quandle of surface links in arbitrary $4$-manifolds. As applications, we extend the work of Sato and Tanaka to show that for any $b \geq 4$ and $m \geq 0$, there exist infinitely many pairwise non-local surface knots with bridge number $b$ in $\mathbb{C}P^2 \#m\overline{\mathbb{C}P^2}$, and we distinguish infinite families of surface knots with isomorphic knot groups, extending results of Tanaka and Taniguchi.