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Abstract
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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 lower bounds on the degeneration of tunnel
number under these operations. In particular, when the graphs are Brunnian
-curves, we show
that the tunnel number is bounded below by the number of prime factors and when the factors
are
-small,
then tunnel number is bounded below by the sum of the tunnel numbers of the
factors. This extends theorems of Scharlemann–Schultens and Morimoto to genus 2
graphs. We are able to prove similar results for the bridge number of such graphs.
The main tool is a family of recently defined invariants for knots, links, and
spatial graphs that detect the unknot and are additive under connected
sum and vertex sum. In this paper, we also show that they detect trivial
-curves.
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Keywords
knot, spatial graphs, tunnel number
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Mathematical Subject Classification
Primary: 57K10, 57K12
Secondary: 57K31
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Milestones
Received: 1 December 2020
Revised: 8 May 2021
Accepted: 27 June 2021
Published: 10 November 2021
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