A tabulation of oriented links

Doll, Helmut; Hoste, Jim

doi:10.1090/S0025-5718-1991-1094946-4

A tabulation of oriented links
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by Helmut Doll and Jim Hoste PDF

Math. Comp. 57 (1991), 747-761 Request permission

Abstract:

In this paper we enumerate all prime, nonsplit, oriented, classical links having two or more components and nine or fewer crossings. Our list is complete up to diffeomorphism of ${S^3}$ and complete reorientation of the link. (That is, reorienting every component of the link.) Previously, only tables of nonoriented links have been compiled. Furthermore, we list, in the case of alternating links, all possible minimal diagrams of each link up to orientation. We also include the skein polynomials of each link. Our methods are direct generalizations of those used by Dowker and Thistlethwaite to enumerate knots. We rely heavily on the HOMFLY and Kauffman polynomials to distinguish inequivalent links. In a few cases these invariants will not suffice and other link invariants are employed. Our table is generated "from scratch" rather than by introducing orientations into already existing nonoriented tables. This provides a check on Conway’s table in the range mentioned above.

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