New examples of tunnel number subadditivity

Details New examples of tunnel number subadditivity New examples of tunnel number subadditivity

2011-11-03

arxiv.org/abs/1111.0980v1

Mathematics

Geometric Topology

14 pages, 6 figures

Scientific paper

If the tunnel number of knot $K$ is denoted $t(K)$, a pair of knots $K_1,K_2$ is said to be subadditive if $t(K_1)+t(K_2)>t(K_1 # K_2)$. We use a slight generalization of the concept of $\mu$-primitivity to construct subadditive pairs of knots of arbitrarily large tunnel number. Also, drawing on a construction of Morimoto, we describe 2-component links of arbitrarily high tunnel number which, in conjunction with certain types of knot, form 3-fold connect sums which asymptotically approach the degeneration ratio 1/3 from above as the tunnel number grows large, with the best example achieving 2/5. From this example we construct knot pairs of a similar type which we conjecture to have the same property.

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