New examples of tunnel number subadditivity (Q1939207)

An unknotting system for a link or a knot \(K\) in the \(3\)-sphere is a collection of pairwise disjoint arcs which meet \(K\) only in their endpoints such that the exterior of the union of \(K\) and the arcs is a handlebody. The minimal number of the arcs over all unknotting systems is called the tunnel number of \(K\), which is denoted by \(t(K)\). Under the connected sum operation \(K_1 \sharp K_2\) of knots or links \(K_1\) and \(K_2\), the sum of tunnel numbers \(t(K_1)\) and \(t(K_2)\) can be preserved (additive), decreased (sub-additive) or increased (super-additive). \textit{K. Morimoto} [Proc. Am. Math. Soc. 123, No.11, 3527--3532 (1995; Zbl 0854.57006)] found a sub-additive pair of knots, and \textit{T. Kobayashi} [J. Knot Theory Ramifications 3, No. 2, 179--186 (1994; Zbl 0818.57006)] showed that \(t(K_1) + t(K_2) - t(K_1 \sharp K_2)\) can be arbitrarily large. \textit{M. Scharlemann} and \textit{J. Schultens} [Math. Ann. 317, No.4, 783--820 (2000; Zbl 0953.57002)] defined the degeneration ratio \(d(K_1, K_2) = 1 - \frac{t(K_1 \sharp K_2)}{t(K_1) + t(K_2)}\), and showed that \(d(K_1, K_2) \leq \frac{3}{5}\). \textit{K. Morimoto} [Geometry and Topology Monographs 12, 265--275 (2007; Zbl 1137.57007)] described some properties that certain knot pairs could have that would allow them to achieve the degeneration ratio of \(\frac{2}{5}\), and recently \textit{J. Nogueira} found knots satisfying these properties in [``On tunnel number degeneration and 2-string tangle decompositions'', Dissertation, UT Austin, (2011)]. The highest known degeneration ratio known for a pair of knots is just \(\frac{2}{5}\). In the present paper, the author constructs a family of links which have the degeneration ratio approaching \(\frac{3}{7}\) when the connected sum is taken with certain knots, and then conjectures that these links can be modified to a family of knots with the same property.