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Some links with non-trivial polynomials and their crossing-numbers

1988; European Mathematical Society; Volume: 63; Issue: 1 Linguagem: Inglês

10.1007/bf02566777

1420-8946

W. B. R. Lickorish, Morwen Thistlethwaite,

Advanced Combinatorial Mathematics

One of the main applications of the Jones polynomial invariant of oriented links has been in understanding links with (reduced, connected) alternating diagrams [2], [8], [9]. The Jones polynomial for such a link is never trivial, and the number of crossings in such a diagram is the crossing-number of the link (that is, no diagram of the link has fewer crossings). Here, the non-triviality of the Jones polynomial is established for a wider class of links that includes most pretzel links and the Whitehead double of any alternating knot. The idea of a semi-alternating link is defined and for such a link the crossing-number is determined. Finally, the crossing-number of any Montesinos link is found. That determination uses both the polynomial of Jones and its two-variable semioriented generalisation (due to Kauffman); it is shown that the latter polynomial is never trivial for a Montesinos knot.