Polynomial invariants of links satisfying cubic skein relations. (English) Zbl 1078.57010

Bellingeri, Paolo; Funar, Louis

Asian J. Math. 8, No. 3, 475-509 (2004).

Two new invariants are defined in this paper. They are closely related, and are determined by two skein relations and their value (1) on the unknot.
One skein relation is the sum of 4 terms, each consisting of a 2-strand braid times some power of the square root of a rational function times a monomial. The other is the sum of 22 terms, each consisting of a 3-strand braid times some power of the square root of the same rational function times a polynomial with up to 6 terms.
The invariants are strong enough to distinguish all knot pairs with up to 10 crossings with the same HOMFLY polynomial. They also determine chirality for all knots with up to 10 crossings for which chirality is not determined by HOMFLY, Kauffman or 2-cabling of HOMFLY. They do not distinguish the mutant Kinoshita-Terasaka and Conway knots.
One of the invariants lies in \(\mathbb Z[\alpha, \beta, (\beta^2 - 2\alpha)^{\pm\epsilon/2}, (\alpha^2 + 2\beta))^{\pm\epsilon/2}]\) modulo a certain polynomial ideal.

Reviewer: Lee P. Neuwirth (Princeton)

MSC:

57M27

Invariants of knots and \(3\)-manifolds (MSC2010)

Keywords:

skein relations; Markov trace; Hecke algebra; quantum invariants; chirality