We present a new theory which describes the collection of all tunnels of tunnel number
knots
in
(up to
orientation-preserving equivalence in the sense of Heegaard splittings) using the disk complex
of the genus–
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
–bridge
knots. The theory extends easily to links, and to allow equivalence of tunnels by
homeomorphisms that may be orientation-reversing.
Keywords
knot, link, tunnel, (1,1), disk complex, two-bridge
