Jones polynomials and classical conjectures in knot theory

Abstract

LETL bea tame link in S3 and VL(f) the Jones polynomial of L defined in [Z]. For a projection E of L, c(L) denotes the number of double points in L and c(L) the minimum number of double points among all projections of L. A link projection t is called proper if L does not contain "removable" double points-. like ,<_: or /.+_,. \/;'I In this paper, we will prove some of the outstanding classical conjectures due to P.G. Tait [7]. THEOREM A. (P. G. Tait Conjecture) 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 non-alternating projection of a prime alternating link cannot be minimal.

Span of the Jones polynomial of an alternating virtual link

“…Kishino [10] Proof By Theorem 1.3, span(D) is not a multiple of four. On the other hand, the span of the f -polynomial of a classical link is a multiple of four [7,13,14]. Thus we have the result.…”

“…For a virtual link L represented by a virtual link diagram D, we define the f -polynomial f L (A) of L by f D (A). The span Theorem 1.1 (Kauffman [7], Murasugi [13], Thistlethwaite [14]) Let L be an alternating link represented by a proper alternating connected link diagram D. Then we have span(L) = 4c(D).…”

A volumish theorem for the Jones polynomial of alternating knots

“…From [Thistlethwaite 1987;Kauffman 1987;Murasugi 1987] Theorem 5.1. Let V K (t) = a n t n + a n+1 t n+1 + · · · + a m t m be the Jones polynomial of an alternating knot K and let G = (V, E) be a checkerboard graph of a reduced alternating projection of K .…”

Parity in knot theory and graph-links

“…Theorem 5.13 (see [180,204]). The span of the Jones-Kauffman polynomial for links with connected shadows and n classical crossings is less than or equal to 4n.…”