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

TL;DR

This paper introduces a new Wirtinger-type presentation for the fundamental quandle of surface links in arbitrary 4-manifolds, enabling advanced distinctions of surface knots beyond the 4-sphere.

Contribution

It extends the fundamental quandle framework to general 4-manifolds using banded unlink diagrams, and constructs infinite families of surface knots with specific bridge numbers.

Findings

Existence of infinitely many non-local surface knots with given bridge numbers in complex projective planes.

Distinction of surface knots with isomorphic knot groups.

Extension of previous work to arbitrary 4-manifolds.

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.