Constructing knot tunnels using giant steps

In 2000, Goda, Scharlemann, and Thompson described a general construction of all tunnels of tunnel number 1 knots using ``tunnel moves''. The theory of tunnels introduced by Cho and McCullough provides a combinatorial approach to understanding tunnel moves. We use it to calculate the numbe...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2010-01, Vol.138 (1), p.375-384
Main Authors: CHO, SANGBUM, MCCULLOUGH, DARRYL
Format: Article
Language:English
Subjects:
Center of gravity
Exact sciences and technology
General mathematics
General, history and biography
Infinite sets
Mathematical induction
Mathematics
Sciences and techniques of general use
Travelers
Triangulation
Tunnels
Vertices
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Summary:In 2000, Goda, Scharlemann, and Thompson described a general construction of all tunnels of tunnel number 1 knots using ``tunnel moves''. The theory of tunnels introduced by Cho and McCullough provides a combinatorial approach to understanding tunnel moves. We use it to calculate the number of distinct minimal sequences of such moves that can produce a given tunnel. As a consequence, we see that for a sparse infinite set of tunnels, the minimal sequence is unique, but generically a tunnel will have many such constructions. Finally, we give a characterization of the tunnels with a unique minimal sequence.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-09-10069-2