Closure of knitted surfaces and surface-links

TL;DR

This paper explores the properties of knitted surfaces and their closures, demonstrating that any surface-link can be represented as a closure of a 2-dimensional knit, and characterizing trivial surface-knots via plat closures.

Contribution

It proves that all surface-links are isotopic to closures of 2-dimensional knits and characterizes trivial surface-knots through plat closures of degree 2 knitted surfaces.

Findings

Any surface-link is isotopic to the closure of a 2-dimensional knit.

Trivial surface-knots are characterized by plat closures of degree 2 knitted surfaces.

Plat closure of degree 2 knitted surfaces yields trivial surface-links.

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.