The combinatorics of knot invariants arising from the study of Macdonald polynomials

This chapter gives an expository account of some unexpected connections which have arisen over the last few years between Macdonald polynomials, invariants of torus knots, and lattice path combinatorics. The study of polynomial knot invariants is a well-known branch of topology which originated in the 1920s with the one-parameter Alexander polynomial [1]. In the early 1980s Jones [41] introduced a different one-parameter polynomial invariant, with important connections to physics. Shortly thereafter a number of authors more or less simultaneously discovered the HOMFLY polynomial, a two-parameter invariant which includes both the Alexander and Jones polynomials as special cases. The HOMFLY polynomial can be calculated recursively through skein relations. In the late 1980s Witten showed that the Jones polynomial and related invariants have an interpretation in terms of Chern–Simons theory, which is central to string theory. In 2006 Dunfield, Gukov, and Rasmussen [14] hypothesized the existence of a three-parameter knot invariant, now known as the “superpolynomial knot invariant” of a knot K, denoted \(\mathcal{P}_{K}(a,q,t)\), which includes the HOMFLY polynomial as a special case. Since then various authors proposed different possible definitions of the superpolynomial, which are conjecturally all equivalent. These definitions typically involve homology though, and are difficult to compute. In the case of torus knots an accepted definition of the superpolynomial has recently emerged from work of Aganagic and Shakirov [6, 7] (using refined Chern–Simons theory) and Cherednik [12] (using the double affine Hecke algebra). Gorsky and Negut [24] showed that these two different constructions yield the same three-parameter knot invariant which is now accepted as the definition of the superpolynomial for torus knots. These constructions involve symmetric functions in a set of variables X known as Macdonald polynomials, which also depend on two extra parameters q, t. These symmetric functions are important in algebraic combinatorics and other areas, and play a central role in various character formulas for S n modules connected to the Hilbert scheme from algebraic geometry. In particular, Haiman’s formula for the bigraded character of the space DH n of diagonal harmonics under the diagonal action of the symmetric group is expressed in terms of Macdonald polynomials. E. Gorsky [22], [47, Appendix] noticed that the coefficient of a j in the superpolynomial of the (n + 1, n) torus knot equals the bigraded multiplicity of a certain hook shape in the character of DH n . This polynomial is known as the (q, t)-Schroder polynomial since the author showed it can be expressed as a weighted sum over Schroder lattice paths in the n × n + 1 rectangle. Gorsky and Negut have shown that the coefficient of a j in the superpolynomial of the (m, n) torus knot can be viewed as the coefficient of a certain hook Schur function in a symmetric function expression involving Macdonald polynomials, and they have derived many explicit identities for this object. In addition Oblomkov, Rasmussen, and Shende [47] have introduced a conjectured extension of the q, t-Schroder polynomial to general (m, n) giving a positive strictly combinatorial expression for the superpolynomial of the (m, n) torus knot. This conjecture connects nicely with an important conjecture in algebraic combinatorics called the rational shuffle conjecture. In the following pages we will describe these developments in more detail.