Differently knotted symplectic surfaces in D^4 bounded by the same transverse knot

TL;DR

This paper demonstrates the existence of two symplectic surfaces in the 4-ball with identical topology and boundary transverse knot, distinguished by the fundamental groups of their complements, highlighting subtle differences in symplectic topology.

Contribution

It introduces the first example of symplectic surfaces with the same boundary knot and topology but different complement fundamental groups.

Findings

Two symplectic surfaces bound the same transverse knot.

The surfaces have the same topology but different complement groups.

Fundamental groups distinguish the surfaces despite identical boundary and topology.

Abstract

In this paper we show that there are two symplectic surfaces in the 4-ball which bound the same transverse knot, have the same topology (as abstract surfaces), and are distinguished by the fundamental groups of their complements.