Twist Spinning Knotted Trivalent Graphs

TL;DR

This paper generalizes the concept of twist spinning from knotted spheres to knotted trivalent graphs, showing that twist spins can be knotted or unknotted depending on the graph, thus extending classical results in higher-dimensional knot theory.

Contribution

It introduces a new concept of knotted 2-dimensional foams as a generalization of knotted spheres and analyzes the knottedness of their twist spins, providing new insights into higher-dimensional knot theory.

Findings

Twist spins of knotted trivalent graphs can be knotted or unknotted.

Defined a new class of objects called knotted 2-dimensional foams.

Identified families of graphs with always unknotted twist spins.

Abstract

In 1965, E. C. Zeeman proved that the (+/-)-twist spin of any knotted sphere in (n-1)-space is unknotted in the n-sphere. In 1991, Y. Marumoto and Y. Nakanishi gave an alternate proof of Zeeman's theorem by using the moving picture method. In this paper, we define a knotted 2-dimensional foam which is a generalization of a knotted sphere and prove that a (+/-)-twist spin of a knotted trivalent graph may be knotted. We then construct some families of knotted graphs for which the (+/-)-twist spins are always unknotted.