Signature, slicing foams, and crossing changes of Klein graphs

TL;DR

This paper extends signature invariants to totally oriented Klein graphs and develops bounds relating signature, slice genus, and unknotting number, providing new tools for analyzing Klein graphs and their transformations.

Contribution

It generalizes Gille-Robert's signature to all totally oriented Klein graphs and establishes bounds linking signature, Euler characteristic, and unknotting number.

Findings

Signature provides lower bounds on orbifold Euler characteristic.

Bounds on unknotting number and Gordian distance for Klein graphs.

Improved unknotting number bounds for certain theta-curves.

Abstract

A totally oriented Klein graph is a trivalent spatial graph in the 3-sphere with a 3-coloring of its edges and an orientation on each bicolored link. A totally oriented Klein foam is a 3-colored 2-complex in the 4-ball whose boundary is a Klein foam and whose bicolored surfaces are oriented. We extend Gille-Robert's signature for 3-Hamiltonian Klein graphs to all totally oriented Klein graphs and develop an analogy of Murasugi's bounds relating the signature, slice genus and unknotting number of knots. In particular, we show that the signature of a totally oriented Klein graph produces a lower bound on the negative orbifold Euler characteristic of certain totally oriented Klein foams bounded by $\Gamma$. When $\Gamma$ is abstractly planar, these negative Euler characteristics, in turn, produce a lower bound on a certain natural unknotting number for $\Gamma$. Mutatis mutandi, we produce lower bounds on the corresponding Gordian distance between two totally oriented Klein graphs that can be related by a sequence of crossing changes. We also give examples of theta-curves for which our lower bounds on unknotting number improve on previously known bounds.