Infinitely many two-variable generalisations of the Alexander–Conway polynomial

De Wit, David¹ ; Ishii, Atsushi ; Links, Jon¹

¹Department of Mathematics, The University of Queensland, 4072 Brisbane, Australia

Algebraic and Geometric Topology, Tome 5 (2005) no. 1, pp. 405-418

Résumé

We show that the Alexander-Conway polynomial Δ is obtainable via a particular one-variable reduction of each two-variable Links–Gould invariant LGm,1, where m is a positive integer. Thus there exist infinitely many two-variable generalisations of Δ. This result is not obvious since in the reduction, the representation of the braid group generator used to define LGm,1 does not satisfy a second-order characteristic identity unless m = 1. To demonstrate that the one-variable reduction of LGm,1 satisfies the defining skein relation of Δ, we evaluate the kernel of a quantum trace.

Détail

DOI : 10.2140/agt.2005.5.405

Keywords: link, knot, Alexander-Conway polynomial, quantum superalgebra, Links–Gould link invariant

Bibliographie

[1] B Abdesselam, D Arnaudon, A Chakrabarti, Representations of Uq(sl(N)) at roots of unity, J. Phys. A 28 (1995) 5495

[2] D De Wit, Automatic evaluation of the Links–Gould invariant for all prime knots of up to 10 crossings, J. Knot Theory Ramifications 9 (2000) 311

[3] D De Wit, An infinite suite of Links–Gould invariants, J. Knot Theory Ramifications 10 (2001) 37

[4] D De Wit, Automatic construction of explicit R matrices for the one-parameter families of irreducible typical highest weight (⋅0m|⋅αn) representations of Uq[gl(m|n)], Comput. Phys. Comm. 145 (2002) 205

[5] D De Wit, J R Links, L H Kauffman, On the Links–Gould invariant of links, J. Knot Theory Ramifications 8 (1999) 165

[6] P Freyd, D Yetter, J Hoste, W B R Lickorish, K Millett, A Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. (N.S.) 12 (1985) 239

[7] M D Gould, J R Links, Y Z Zhang, Type–I quantum superalgebras, q–supertrace, and two-variable link polynomials, J. Math. Phys. 37 (1996) 987

[8] A Ishii, The Links–Gould invariant of closed 3–braids, J. Knot Theory Ramifications 13 (2004) 41

[9] A Ishii, Algebraic links and skein relations of the Links–Gould invariant, Proc. Amer. Math. Soc. 132 (2004) 3741

[10] A Ishii, The Links–Gould polynomial as a generalization of the Alexander–Conway polynomial, Pacific J. Math. 225 (2006) 273

[11] A Ishii, T Kanenobu, Different links with the same Links–Gould invariant, Osaka J. Math. 42 (2005) 273

[12] V Kac, Representations of classical Lie superalgebras, from: "Differential geometrical methods in mathematical physics, II (Proc. Conf., Univ. Bonn, Bonn, 1977)", Lecture Notes in Math. 676, Springer (1978) 597

[13] L H Kauffman, H Saleur, Free fermions and the Alexander–Conway polynomial, Comm. Math. Phys. 141 (1991) 293

[14] J R Links, M D Gould, Two variable link polynomials from quantum supergroups, Lett. Math. Phys. 26 (1992) 187

[15] J R Links, M D Gould, R B Zhang, Quantum supergroups, link polynomials and representation of the braid generator, Rev. Math. Phys. 5 (1993) 345

[16] T Ohtsuki, Quantum invariants, 29, World Scientific Publishing Co. (2002)

[17] T D Palev, N I Stoilova, J Van Der Jeugt, Finite-dimensional representations of the quantum superalgebra Uq[gl(n∕m)] and related q–identities, Comm. Math. Phys. 166 (1994) 367

[18] R B Zhang, Finite-dimensional irreducible representations of the quantum supergroup Uq(gl(m∕n)), J. Math. Phys. 34 (1993) 1236

[19] R B Zhang, Quantum supergroups and topological invariants of three-manifolds, Rev. Math. Phys. 7 (1995) 809