Infinitely many pairs of spatial surfaces

TL;DR

This paper constructs infinitely many pairs of Seifert surfaces for each link in the 3-sphere, sharing certain invariants but not being ambiently isotopic, thereby advancing the understanding of spatial surface classification.

Contribution

It introduces a method to generate infinitely many non-isotopic Seifert surfaces with identical invariants for each link, using algebraic invariants derived from multiple group racks.

Findings

Constructed infinitely many pairs of Seifert surfaces for each link.

Identified invariants that distinguish non-isotopic surfaces.

Showed that surfaces can share invariants yet differ topologically.

Abstract

A multiple group rack (MGR) is an algebraic system which is used to construct invariants of spatial surfaces, which are compact surfaces embedded in the $3$-sphere $S^{3}$. Seifert surfaces for links are spatial surfaces. In this paper, we present an infinitely many pairs of Seifert surfaces for each link, where each pair satisfies the following condtions: (i) their regular neighborhoods in $S^{3}$ are ambiently isotopic, (ii) their Seifert matrices are unimodularly congruent, and (iii) the two Seifert surfaces are not ambiently isotopic. In order to prove (iii), we distinguish the Seifert surfaces using the above invariants.