On the classification of knots

Perko, Kenneth A.

doi:10.1090/S0002-9939-1974-0353294-X

On the classification of knots
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by Kenneth A. Perko

Proc. Amer. Math. Soc. 45 (1974), 262-266

DOI: https://doi.org/10.1090/S0002-9939-1974-0353294-X

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Abstract:

Linking numbers between branch curves of irregular covering spaces of knots are used to extend the classification of knots through ten crossings and to show that the only amphicheirals in Reidemeister’s table are the seven identified by Tait in 1884. Diagrams of the 165 prime $10$-crossing knot types are appended. (Murasugi and the author have proven them prime; Conway claims proof that the tables are complete.) Including the trivial type, there are precisely 250 prime knots with ten or fewer crossings.

References

Gerhard Burde, Darstellungen von Knotengruppen und eine Knoteninvariante, Abh. Math. Sem. Univ. Hamburg 35 (1970), 107–120 (German). MR 276956, DOI 10.1007/BF02992480
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
R. H. Fox, Metacyclic invariants of knots and links, Canadian J. Math. 22 (1970), 193–201. MR 261584, DOI 10.4153/CJM-1970-025-9

Non-alternate $\pm$ knots

39

Kunio Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965), 387–422. MR 171275, DOI 10.1090/S0002-9947-1965-0171275-5

Knotentheorie

Horst Schubert, Knoten mit zwei Brücken, Math. Z. 65 (1956), 133–170 (German). MR 82104, DOI 10.1007/BF01473875

The first seven orders of knottiness

32

Tenfold knottiness

32