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- W2950594595 abstract "This is the third of three papers that refine and extend portions of our earlier preprint, depth of a knot tunnel. Together, they rework the entire preprint. In this paper, we use the theory of tunnel number 1 knots that we introduced in tree of knot to strengthen the Tunnel Leveling Theorem of H. Goda, M. Scharlemann, and A. Thompson. This yields considerable information about bridge numbers of tunnel number 1 knots. In particular, we calculate the minimum bridge number of a knot as a function of the maximum depth invariant d of its tunnels. The growth of this value is on the order of (1+sqrt{2})^d. We also find the maximum bridge number as a function of the number of cabling constructions needed to produce the tunnel, showing in particular that the maximum bridge number of a knot produced by n cabling constructions is the (n+2)nd Fibonacci number. Finally, we examine the special case of the middle tunnels of torus knots." @default.
- W2950594595 created "2019-06-27" @default.
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- W2950594595 date "2008-12-07" @default.
- W2950594595 modified "2023-09-27" @default.
- W2950594595 title "Tunnel leveling, depth, and bridge numbers" @default.
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