Surfaces in $D^4$ with the same boundary and fundamental group

TL;DR

This paper constructs pairs of non-isotopic symplectic surfaces in the 4-disk with identical boundary knots and isomorphic complement fundamental groups, highlighting subtle differences in symplectic topology.

Contribution

It introduces a method to produce non-isotopic symplectic surfaces sharing the same boundary and fundamental group, expanding understanding of surface distinctions in 4-manifolds.

Findings

Existence of non-isotopic symplectic surfaces with same boundary and fundamental group

Explicit construction of such surface pairs

Connection between braided surfaces and symplectic surfaces in 4-disk

Abstract

We construct a family of pairs of non-isotopic symplectic surfaces in the standard symplectic $4$-disk such that they are bounded by the same transverse knot in the standard contact $3$-sphere and fundamental groups of their complements are isomorphic. In the appendix, we prove explicitly that one can obtain a symplectic surface in the standard symplectic $4$-disk from a braided surface in a bidisk.