Infinitely many pairs of spatial surfaces

TL;DR

The paper addresses distinguishing Seifert surfaces for a fixed link by constructing infinite families of surface pairs with identical regular neighborhoods and unimodularly congruent Seifert matrices, yet not ambiently isotopic. It achieves this by combining Seifert-matrix invariants with coloring invariants from multiple group racks (MGRs), leveraging diagrammatic moves for spatial surfaces and handlebody-knots. The main result is an explicit infinite collection of surface pairs that are topologically indistinguishable via neighborhoods or Seifert matrices but distinguishable via MGR colorings, illustrating deeper complexity in spatial surface classification. This advances invariant-based discrimination in the study of spatial surfaces and handlebody-knots, providing concrete methods to distinguish non-equivalent Seifert surfaces sharing the same boundary.

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.

Figures (9)

• Figure 1: A process of constructing a spatial surface from a diagram
• Figure 2: Local moves on diagrams of spatial trivalent graphs
• Figure 3: All orientations around trivalent vertices
• Figure 4: $X$-coloring conditions
• Figure 5: A diagram $D$ of the Seifert surface $F$ for $L$

Theorems & Definitions (2)

• proof
• proof