Jones polynomials and classical conjectures in knot theory
The paper contains proofs of several of the classical conjectures of P. G. Tait on knot projections. A regular projection \(\tilde L\) of a link \(L\subset S^ 3\) is called proper, if there are no evident ``flips'' reducing the number of double points. Theorem A: Two (connected and proper) alternating projections of an alternating link have the same number of double points. Theorem B: The minimal projection of an alternating link is alternating. In other words, an alternating link always has an alternating projection that has the minimum number of double points among all projections. Moreover, a nonalternating projection of a prime alternating link cannot be minimal. In addition, a corollary is proved: If a knot K is alternating and amphicheiral, then any proper alternating projection has an even number of double points. Furthermore, the twist number (writhe) is shown to be an invariant of proper special alternating projections. (The author and others have since shown that the restriction ``special'' can be dropped.) The proof establishes a close connection between the minimal number of double points of a projection \(\tilde L\) and the degree of Jones polynomial \(V_ L(t)\).
scientific article; zbMATH DE number 4022163