On the Hecke algebras and the colored HOMFLY polynomial

By Xiaoxia Lin, Haocong Zheng. 2006

120 citations recorded by the source corpus.

The colored HOMFLY polynomial is the quantum invariant of oriented links in S 3 associated with irreducible representations of the quantum group U q (sl N ). In this paper, using an approach to calculate quantum invariants of links via the cabling-projection rule, we derive a formula for the colored HOMFLY polynomial in terms of the characters of the Hecke algebras and Schur polynomials. The technique leads to a fairly simple formula for the colored HOMFLY polynomial of torus links. This formula allows us to test the Labastida-Marino-Vafa conjecture, which reveals a deep relationship between Chern-Simons gauge theory and string theory, on torus links.

Subfields: Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/3c7b024c2cf08c0df44626cf03fd707445349e3d

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In Geometry and Topology

1. Marcos Mariño
2. Sergei Gukov
3. B. Eynard
4. Marko Stosic
5. Andrea Brini
6. E. Witten
7. C. Vafa
8. Pavel Putrov

In Nuclear and High Energy Physics

1. Marcos Mariño
2. B. Eynard
3. Sergei Gukov
4. Andrea Brini
5. Marko Stosic
6. C. Vafa
7. E. Witten
8. A. Klemm

Citations in the corpus

• Torus Knots and Mirror Symmetry (2011, methodology): “This is precisely the formula obtained by Rosso and Jones in [46] (see also [37] for a more transparent phrasing).”
• Representations of Rational Cherednik algebras with minimal support and torus knots (2013, unknown): “or1] and references therein for more details on Adams operations. Definition 4.6. Let us define the coefficients cµ λ,n0 by the equation (8) Ψn0(sλ) = X |µ|=n0|λ| cµ λ,n0 sµ. Theorem 4.7. ([RJ], see also [LZ],[Ste]) The HOMFLY polynomial of the λ-colored (m0,n0) torus knot can be computed using the form”
• Homological algebra of knots and BPS states (2011, background): “ superpolynomial for many small knots. As the first example, we consider the trefoil knot: Example. The trefoil knot 31 The reduced -colored HOMFLY polynomial of the trefoil knot is equal to (see e.g. [26, 57]): P (31) = a2q−2 +a2q +a2q2 +a2q4 −a3 −a3q −a3q3 −a3q4 +a4q3 . The homology Hso(3),V (3 1) ”
• Super $q$-Howe duality and web categories (2015, unknown): “Up to different conventions, this is shown for example in [17, Corollary 4.5].”