The colored Jones polynomials and the simplicial volume of a knot. (English) Zbl 0983.57009
Using the quantum dilogarithm, R. M. Kashaev defined a family of complex-valued link invariants indexed by integers \(N \geq 2\) [A link invariant from quantum dilogarithm, Mod. Phys. Lett. A 10, No. 19, 1409-1418 (1995; Zbl 1022.81574)]; after verifying it for the three simplest hyperbolic knots, he conjectured that for any hyperbolic knot the invariant grows as \(\text{exp(Vol}(K)N/2\pi\)) when \(N\) goes to infinity, where \(\text{Vol}(K)\) denotes the volume of the complement of the knot. In the present paper, to “reveal his mysterious definition” and to make this intriguing conjecture more accessible, it is shown that his invariant is nothing but a specialization of the colored Jones polynomial (each component of the link being decorated with an irreducible representation of the Lie algebra \(\text{sl}(2,\mathbb C)\)). Whereas the original Jones polynomial corresponds to the case that all the colors are identical to the 2-dimensional fundamental representation, it is shown that Kashaev’s invariant with parameter \(N\) coincides with the colored Jones polynomial, in a certain normalization, with every color the \(N\)-dimensional representation evaluated at the primitive \(N\)th root of unity.
It is also shown that the generalized multivariable Alexander polynomial as defined by Y. Akutsu, T. Deguchi and T. Ohtsuki [J. Knot Theory Ramifications 1, No. 2, 161-184 (1992; Zbl 0758.57004)], parametrized by an integer \(N\) and complex numbers decorating the components, becomes equal to Kashaev’s invariant when all colors are chosen to be \((N-1)/2\) (by showing that their representation becomes the usual representation corresponding to the irreducible \(N\)-dimensional representation of \(\text{sl}(2,\mathbb C)\)). Finally, observing that Gromov’s simplicial volume is additive and unchanged by mutation, it is conjectured that Kashaev’s invariants (= specializations of the colored Jones polynomial = specializations of the generalized multivariable Alexander polynomial) determine the simplicial volume (“dröm i Djursholm”: “if our dream comes true, then we can show that a knot is trivial if and only if all of its Vassiliev invariants are trivial”).
It is also shown that the generalized multivariable Alexander polynomial as defined by Y. Akutsu, T. Deguchi and T. Ohtsuki [J. Knot Theory Ramifications 1, No. 2, 161-184 (1992; Zbl 0758.57004)], parametrized by an integer \(N\) and complex numbers decorating the components, becomes equal to Kashaev’s invariant when all colors are chosen to be \((N-1)/2\) (by showing that their representation becomes the usual representation corresponding to the irreducible \(N\)-dimensional representation of \(\text{sl}(2,\mathbb C)\)). Finally, observing that Gromov’s simplicial volume is additive and unchanged by mutation, it is conjectured that Kashaev’s invariants (= specializations of the colored Jones polynomial = specializations of the generalized multivariable Alexander polynomial) determine the simplicial volume (“dröm i Djursholm”: “if our dream comes true, then we can show that a knot is trivial if and only if all of its Vassiliev invariants are trivial”).
Reviewer: Bruno Zimmermann (Trieste)
MSC:
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
| 57M50 | General geometric structures on low-dimensional manifolds |
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
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