Sangbum Cho

The tree of knot tunnels

We present a new theory which describes the collection of all tunnels of tunnel number 1 knots in the 3-sphere (up to orientation-preserving equivalence in the sense of Heegaard splittings) using the disk complex of the genus-2 handlebody and associated structures. It shows that each knot tunnel is obtained from the tunnel of the trivial knot by a uniquely determined sequence of simple cabling constructions. A cabling construction is determined by a single rational parameter, so there is a corresponding numerical parameterization of all tunnels by sequences of such parameters and some additional data. Up to superficial differences in definition, the final parameter of this sequence is the Scharlemann-Thompson invariant of the tunnel, and the other parameters are the Scharlemann-Thompson invariants of the intermediate tunnels produced by the constructions. We calculate the parameter sequences for tunnels of 2-bridge knots. The theory extends easily to links, and to allow equivalence of tunnels by homeomorphisms that may be orientation-reversing.

Arc distance equals level number

Let K be a knot in 1-bridge position with respect to a genus-g Heegaard surface that splits a 3-manifold M into two handlebodies V and W. One can move K by isotopy keeping K∩V in V and K∩W in W so that K lies in a union of n parallel genus-g surfaces tubed together by n ― 1 straight tubes, and K intersects each tube in two arcs connecting the ends. We prove that the minimum n for which this is possible is equal to a Hempel-type distance invariant defined using the arc complex of the two-holed genus-g surface.

The disk complex and 2-bridge knots

We give an alternative proof of a result of Kobayashi and Saeki that every genus one

Cabling sequences of tunnels of torus knots

1

Tunnel leveling, depth, and bridge numbers

-bridge position of a non-trivial

CONSTRUCTING KNOT TUNNELS USING GIANT STEPS

2

Connected Primitive Disk Complexes and Genus Two Goeritz Groups of Lens Spaces

-bridge knot is a stabilization.