Abstract
Recent work of the first author, Negut and Rasmussen, and of Oblomkov and Rozansky in the context of Khovanov--Rozansky knot homology produces a family of polynomials in $q$ and $t$ labeled by integer sequences. These polynomials can be expressed as equivariant Euler characteristics of certain line bundles on flag Hilbert schemes. The $q,t$-Catalan numbers and their rational analogues are special cases of this construction. In this paper, we give a purely combinatorial treatment of these polynomials and show that in many cases they have nonnegative integer coefficients. For sequences of length at most 4, we prove that these coefficients enumerate subdiagrams in a certain fixed Young diagram and give an explicit symmetric chain decomposition of the set of such diagrams. This strengthens results of Lee, Li and Loehr for $(4,n)$ rational $q,t$-Catalan numbers.
Highlights
The last decade revealed deep, and yet partially conjectural connections [11, 9, 12, 13, 6, 7, 8] of the HOMFLY-PT link homologies with various intricate constructions in algebraic combinatorics such as q, t-Catalan numbers of Garsia and Haiman [4], LLT polynomials [14], and the elliptic Hall algebra [25]
For sequences of length at most 4, we prove that these coefficients enumerate subdiagrams in a certain fixed Young diagram and give an explicit symmetric chain decomposition of the set of such diagrams
Motivated by the geometry of the flag Hilbert scheme of points on the plane we can approximate the invariants of such knots with the following combinatorial expressions
Summary
The last decade revealed deep, and yet partially conjectural connections [11, 9, 12, 13, 6, 7, 8] of the HOMFLY-PT link homologies with various intricate constructions in algebraic combinatorics such as q, t-Catalan numbers of Garsia and Haiman [4], LLT polynomials [14], and the elliptic Hall algebra [25]. A priori, this is a rational function in q and t, but we prove in Section 2.3 that it is always a polynomial in q and t with integer coefficients. This polynomial can be expressed as a sum over Tesler matrices with row sums ai as in [9] and especially [1], where similar polynomials have already appeared.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
