Tunnel number and bridge number of composite genus 2 spatial graphs

TL;DR

This paper investigates how the tunnel number and bridge number of genus 2 spatial graphs behave under operations like connected sum and vertex sum, establishing bounds and extending knot theory results to spatial graphs.

Contribution

It provides sharp bounds for the tunnel number and bridge number of composite genus 2 spatial graphs, extending classical knot theory results to more complex spatial graphs.

Findings

Tunnel number bounds for composite genus 2 graphs

Bridge number lower bounds for Brunnian $ heta$-curves

Extension of Morimoto's theorem to genus 2 spatial graphs

Abstract

Connected sum and trivalent vertex sum are natural operations on genus 2 spatial graphs and, as with knots, tunnel number behaves in interesting ways under these operations. We prove sharp Scharlemann-Schultens type bounds for the tunnel number of a composite genus 2 spatial graph. For the tunnel number of a composite Brunnian $\theta$-curve, our result implies that the tunnel number is at least the number of summands, as in the knot case. We also prove a version of a theorem of Morimoto for knots: the tunnel number of a composite m-small genus 2 spatial graph is at least the sum of the tunnel numbers of the factors. We also study lower bounds for the bridge number of composite genus 2 graphs. In particular, our results imply that for a Brunnian composite $\theta$-curve having $m$ factors in its prime factorization, the bridge number is at least $m+3/2$.