Additive invariants for knots, links and graphs in 3-manifolds

TL;DR

This paper introduces two new additive invariants for knots, links, and graphs in 3-manifolds that detect the unknot and relate to bridge and tunnel numbers, with applications to knot sum properties.

Contribution

It defines novel invariants for (3-manifold, graph) pairs that are additive and detect the unknot, extending Gabai's width and relating to key knot invariants.

Findings

Invariants detect the unknot.

Invariants are additive under connected sum.

Applications to tunnel number and bridge number.

Abstract

We define two new families of invariants for (3-manifold, graph) pairs which detect the unknot and are additive under connected sum of pairs and (-1/2)-additive under trivalent vertex sum of pairs. The first of these families is closely related to both bridge number and tunnel number. The second of these families is a variation and generalization of Gabai's width for knots in the 3-sphere. We give applications to the tunnel number and higher genus bridge number of connected sums of knots.