1991/01/01 by Helmut Doll, Jim Hoste · 1 citation
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Table (database) #Annotation #Mathematics #Link (geometry) #Prime (order theory) #Type (biology) #Computer science #Algorithm #Combinatorics #Artificial intelligence #Data mining
paper · pdf · doi:10.1090/s0025-5718-1991-1094946-4
openalex publication_date 1991/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/27
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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S cubed"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">S3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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, <italic>all possible minimal diagrams</italic> 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.