Closure of knitted surfaces and surface-links

TL;DR

The paper develops a combinatorial and geometric framework for representing surface-links in $\mathbb{R}^4$ via knitted surfaces, extending the notion of braids to $n$-knits and their traces. It proves an Alexander-type theorem: every orientable surface-link is ambient isotopic to the closure of some 2-dimensional knit, and introduces plat closures for both braids and braided surfaces, showing that plat closures of degree-$2$ knits yield trivial surface-links and that every trivial surface-link arises as such a plat closure. The work relies on motion-picture methods, band-surgery along bands, and a chart description (2-charts) to encode degree-2 knits, culminating in a normal form and index notions (knit plat index, knit index) that relate to existing braid-based invariants. The results provide a concrete, diagrammatic approach to classifying and constructing surface-links in four dimensions, with explicit handling of the degree-$2$ case and corollaries for trivial links and index bounds. Overall, the paper advances a unified description of surface-links through knitted surfaces and their closures, offering practical tools for identifying and manipulating 4D links via 2D combinatorial data.

Abstract

A knitted surface is a surface with or without closed components smoothly properly embedded in $D^2 \times B^2$, which is a generalization of a braided surface. A knitted surface is called a 2-dimensional knit if its boundary is the closure of a trivial braid. From a 2-dimensional knit $S$, we obtain a surface-link in $\mathbb{R}^4$ by taking the closure of $S$. We show that any surface-link is ambient isotopic to the closure of some 2-dimensional knit. Further, we consider another type of the closure of a knitted surface, called the plat closure. It is known that any trivial surface-knot is ambient isotopic to the plat closure of a knitted surface of degree 2. We show that the plat closure of any knitted surface of degree 2 is a trivial surface-link, and any trivial surface-link is ambient isotopic to the plat closure of a knitted surface of degree $2$. We also show the same result for the closure of 2-dimensional knits of degree 2.

Figures (18)

• Figure 1: Generators $\sigma_i$, $\sigma_i^{-1}$ and $\tau_i$ of the monoid $D_n$ of $n$-knits.
• Figure 2: Motion pictures of the knitted surfaces presented by (1) $e \leftrightarrow \sigma_i$, and (2) $e \leftrightarrow \tau_i$, where we omit the $j$th strings for $j \neq i, i+1$. This figure is NY.
• Figure 3: Motion picture of the knitted surface presented by $\tau_i \leftrightarrow \tau_i\tau_i$, where we omit the $j$th strings for $j \neq i, i+1$. This figure is similar to NY.
• Figure 4: The plat closure of a braid.
• Figure 5: Adequate braids which generate the Hilden subgroup $K_4$: from the left, $\sigma_{1}$, $\sigma_3$, $\sigma_{2}\sigma_{1}\sigma_{3}\sigma_{2}$, and $\sigma_{2}\sigma_{1}\sigma^{-1}_{3}\sigma^{-1}_{2}$. This figure is NY.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Definition 2.1
• Theorem 3.1: Kamada94-2
• Theorem 3.2: Yasuda21
• Definition 4.1
• Definition 4.2
• Definition 5.1
• Proposition 5.2