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On the Links–Gould invariant and the square of the Alexander polynomial

Ben-Michael Kohli

IMB UMR5584, CNRS, Université Bourgogne Franche-Comté, F-21000 Dijon, France

E-mail Address: Ben-Michael.Kohli@u-bourgogne.fr

https://doi.org/10.1142/S0218216516500061Cited by:3 (Source: Crossref)

Abstract

This paper gives a connection between well-chosen reductions of the Links–Gould invariants of oriented links and powers of the Alexander–Conway polynomial. This connection is obtained by showing the representations of the braid groups we derive the specialized Links–Gould polynomials from can be seen as exterior powers of a direct sum of Burau representations.

Keywords:

AMSC: 57M27, 17B37

References

1. D. De Wit, An infinite suite of Links–Gould invariants, J. Knot Theory Ramifications 10(1) (2001) 37–62, MR 2002e:57015 Link, Web of Science, Google Scholar
2. D. De Wit, A. Ishii and J. Links, Infinitely many two-variable generalisations of the Alexander–Conway polynomial, Algebr. Geom. Topol. 5 (2005) 405–418. Crossref, Web of Science, Google Scholar
3. D. De Wit, L. H. Kauffman and J. Links, On the Links–Gould invariant of links, J. Knot Theory Ramifications 8(2) (1999) 165–199, MR 2000j:57020. Link, Web of Science, Google Scholar
4. R. H. Fox and G. Torres, Dual presentations of the group of a knot, Ann. of Math. (2) 59(2) (1954) 211–218. Crossref, Web of Science, Google Scholar
5. N. Geer and B. Patureau-Mirand, Multivariable link invariants arising from $s l (2 | 1)$ and the Alexander polynomial, J. Pure Appl. Algebra 210(1) (2007) 283–298. Crossref, Web of Science, Google Scholar
6. M. D. Gould, P. S. Isaac and J. L. Werry, Matrix elements for type 1 unitary irreducible representations of the Lie superalgebra $g l (m | n)$ , J. Math. Phys. 55 (2014) 011703, doi: 10.1063/1.4861706. Crossref, Web of Science, Google Scholar
7. A. Ishii, The Links–Gould polynomial as a generalization of the Alexander–Conway polynomial, Pacific J. Math. 225(2) (2006) 273–285. Crossref, Web of Science, Google Scholar
8. S. M. Khoroshkin and V. N. Tolstoy, Universal R-matrix for quantized (super) algebras, Comm. Math. Phys. 141 (1991) 599–617. Crossref, Web of Science, Google Scholar
9. W. B. R. Lickorish, An Introduction to Knot Theory, Graduate Texts in Mathematics, Vol. 175 (Springer, 1997). Crossref, Google Scholar
10. J. Links and M. D. Gould, Two variable link polynomials from quantum supergroups, Lett. Math. Phys. 26(3) (1992) 187–198. Crossref, Web of Science, Google Scholar
11. I. Marin and E. Wagner, A cubic defining algebra for the Links–Gould polynomial, Adv. Math. 248 (2013) 1332–1365. Crossref, Web of Science, Google Scholar
12. T. Ohtsuki, Quantum Invariants. A Study of Knots, 3-Manifolds, and their Sets, Series on Knots and Everything, Vol. 29 (World Scientific, 2002). Google Scholar
13. N. Reshetikhin, C. Stroppel and B. Webster, Schur–Weyl-Type duality for quantized $g l (1 | 1)$ , the Burau representation of braid groups, and invariants of tangled graphs, eds. Perspectives in Analysis, Geometry, and Topology: On the Occasion of the 60th Birthday of Oleg Viro, eds. I. Itenberg et al., Progress in Mathematics, Vol. 296 (Springer, 2011). Google Scholar
14. O. Viro, Quantum relatives of the Alexander polynomial, St. Petersburg Math. J. 18(3) (2007) 391–457. Crossref, Google Scholar
15. H. Yamane, Quantized enveloping Algebras associated with simple Lie superalgebras and their universal R-matrices, Publ. Res. Inst. Math. Sci. 30 (1994) 15–87. Crossref, Web of Science, Google Scholar