Abstract
We present a short and unified representation-theoretical treatment of type A link invariants (that is, the HOMFLY–PT polynomials, the Jones polynomial, the Alexander polynomial and, more generally, the quantum \(\mathfrak {gl}_{m|n}\) invariants) as link invariants with values in the quantized oriented Brauer category.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
Note that to precisely match the definitions, one should replace q by \(q^{-1}\) and \(q^{\beta }\) by \(q^{-\beta }\).
References
A.K. Aiston, H.R. Morton, Idempotents of Hecke algebras of type A. J. Knot Theory Ramif. 7(4), 463–487 (1998). https://doi.org/10.1142/S0218216598000243
J.W. Alexander, Topological invariants of knots and links. Trans. Am. Math. Soc. 30(2), 275–306 (1928). https://doi.org/10.2307/1989123
V. Chari, A. Pressley, A Guide to Quantum Groups (Cambridge University Press, Cambridge, 1994)
J. Comes, B. Wilson, Deligne’s category \(\underline{\rm {REP}}(GL_\delta )\) and representations of general linear supergroups. Represent. Theory 16, 568–609 (2012). https://doi.org/10.1090/S1088-4165-2012-00425-3
R. Dipper, S. Doty, F. Stoll, The quantized walled Brauer algebra and mixed tensor space. Algebras Represent. Theory 17(2), 675–701 (2014). https://doi.org/10.1007/s10468-013-9414-2
P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millett, A. Ocneanu, A new polynomial invariant of knots and links. Bull. Am. Math. Soc. (N.S.) 12(2), 239–246 (1985). https://doi.org/10.1090/S0273-0979-1985-15361-3
N. Geer, B. Patureau-Mirand, V. Turaev, Modified quantum dimensions and re-normalized link invariants. Compos. Math. 145(1), 196–212 (2009). https://doi.org/10.1112/S0010437X08003795
E. Gorsky, S. Gukov, M. Stosic, Quadruply-graded colored homology of knots (2013), arXiv:1304.3481
A. Gyoja, A q-analogue of Young symmetrizer. Osaka J. Math. 23(4), 841–852 (1986), http://projecteuclid.org/euclid.ojm/1200779724
V.F.R. Jones, A polynomial invariant for knots via von Neumann algebras. Bull. Am. Math. Soc. (N.S.) 12(1), 103–111 (1985). https://doi.org/10.1090/S0273-0979-1985-15304-2
C. Kassel, Quantum Groups. Graduate Texts in Mathematics, vol. 155 (Springer, New York, 1995)
A. Mathas, Iwahori-Hecke Algebra and Schur Algebras of the Symmetric Group. University Lecture Series, vol. 15 (American Mathematical Society, Providence, 1999)
T. Ohtsuki, Quantum Invariants. Series on Knots and Everything, vol. 29 (World Scientific, Singapore, 2002)
J.H. Przytycki, P. Traczyk, Conway algebras and skein equivalence of links. Proc. Am. Math. Soc. 100(4), 744–748 (1987). https://doi.org/10.2307/2046716
H. Queffelec, A. Sartori, Mixed quantum skew Howe duality and link invariants of type A. J. Pure Appl. Algebra. 223(7), 2733–2779 (2019) . https://doi.org/10.1016/j.jpaa.2018.09.014
N.Y. Reshetikhin, V.G. Turaev, Ribbon graphs and their invariants derived from quantum groups. Commun. Math. Phys. 127(1), 1–26 (1990), http://projecteuclid.org/euclid.cmp/1104180037
A. Sartori, The Alexander polynomial as quantum invariant of links. Ark. Mat. 53(1), 177–202 (2015). https://doi.org/10.1007/s11512-014-0196-5
D. Tubbenhauer, P. Vaz, P. Wedrich, Super q-howe duality and web categories. Algebr. Geom. Topol. 17(6), 3703–3749 (2017). https://doi.org/10.2140/agt.2017.17.3703
Acknowledgements
H.Q. was funded by the ARC DP 140103821.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Queffelec, H., Sartori, A. (2019). A Note on \(\mathfrak {gl}_{m|n}\) Link Invariants and the HOMFLY–PT Polynomial. In: Adams, C., et al. Knots, Low-Dimensional Topology and Applications. KNOTS16 2016. Springer Proceedings in Mathematics & Statistics, vol 284. Springer, Cham. https://doi.org/10.1007/978-3-030-16031-9_13
Download citation
DOI: https://doi.org/10.1007/978-3-030-16031-9_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-16030-2
Online ISBN: 978-3-030-16031-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)