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The Structure and Enumeration of Link Projections
Author(s):
Martin
Bridgeman
Journal:
Trans. Amer. Math. Soc.
348
(1996),
2235-2248.
MSC (1991):
Primary 57M25, 57M15, 05C30, 05C85;
Secondary 53A35
MathSciNet review:
1321569
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Abstract:
We define a decomposition of link projections whose pieces we call atoroidal graphs. We describe a surgery operation on these graphs and show that all atoroidal graphs can be generated by performing surgery repeatedly on a family of well-known link projections. This gives a method of enumerating atoroidal graphs and hence link projections by recomposing the pieces of the decomposition.
References:
- [B]
- M. Bridgeman, Volume Increase under Dehn Drilling Operations, Phd. thesis, Princeton, June 1994.
- [C]
- J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra, Pergamon, Oxford, (1970). MR 41:2661
- [K1]
- T. P. Kirkman, The enumeration, description and construction of knots of fewer than ten crossings, Trans. Roy. Soc. Edinburgh, 32 (1885).
- [K2]
- T. P. Kirkman, The 364 unifilar knots of ten crossings enumerated and defined, Trans. Roy. Soc. Edinburgh, 32 (1885).
- [L1]
- C. N. Little, Non-alternate
 knots, of order eight and nine, Trans. Roy. Soc. Edinburgh, 35 (1889). - [L2]
- C. N. Little, Alternate
 knots of order 11, Trans. Roy. Soc. Edinburgh, 36 (1890). - [T]
- P. G. Tait, On knots I, II, III (1887, 1884, 1885), Scientific Papers I.
- [Th]
- W. P. Thurston, The geometry and topology of three manifolds, Princeton Lecture Notes, (1979).
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Additional Information:
Martin
Bridgeman
Affiliation:
Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, California 94720
Address at time of publication:
Department of Mathematics, Loyola University, New Orleans, Louisiana 70118
Email:
bridgemn@beta.loyno.edu
DOI:
10.1090/S0002-9947-96-01484-5
PII:
S 0002-9947(96)01484-5
Received by editor(s):
October 15, 1994
Additional Notes:
Research at MSRI is supported in part by NSF grant no. DMS-9022140
Copyright of article:
Copyright
1996,
American Mathematical Society
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