The Structure and Enumeration of Link Projections

Bridgeman, Martin

doi:10.1090/S0002-9947-96-01484-5

The Structure and Enumeration of Link Projections
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by Martin Bridgeman PDF

Trans. Amer. Math. Soc. 348 (1996), 2235-2248 Request permission

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

M. Bridgeman, Volume Increase under Dehn Drilling Operations, Phd. thesis, Princeton, June 1994.
J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) Pergamon, Oxford, 1970, pp. 329–358. MR 0258014
T. P. Kirkman, The enumeration, description and construction of knots of fewer than ten crossings, Trans. Roy. Soc. Edinburgh, 32 (1885).
T. P. Kirkman, The 364 unifilar knots of ten crossings enumerated and defined, Trans. Roy. Soc. Edinburgh, 32 (1885).
C. N. Little, Non-alternate $\pm$ knots, of order eight and nine, Trans. Roy. Soc. Edinburgh, 35 (1889).
C. N. Little, Alternate $\pm$ knots of order 11, Trans. Roy. Soc. Edinburgh, 36 (1890).
P. G. Tait, On knots I, II, III (1887, 1884, 1885), Scientific Papers I.
W. P. Thurston, The geometry and topology of three manifolds, Princeton Lecture Notes, (1979).