Categorifying fractional Euler characteristics, Jones–Wenzl projectors and $3j$-symbols

Abstract

Abstract. We study the representation theory of the smallest quantum group and its categorification. The first part of the paper contains an easy visualization of the 3j-symbols in terms of weighted signed line arrangements in a fixed triangle and new binomial expressions for the 3j-symbols. All these formulas are realized as graded Euler characteristics. The 3j-symbols appear as new generalizations of Kazhdan-Lusztig polynomials.A crucial result of the paper is that complete intersection rings can be employed to obtain rational Euler characteristics, hence to categorify rational quantum numbers. This is the main tool for our categorification of the Jones-Wenzl projector, Θ-networks and tetrahedron networks. Networks and their evaluations play an important role in the Turaev-Viro construction of 3-manifold invariants. We categorify these evaluations by Ext-algebras of certain simple Harish-Chandra bimodules. The relevance of this construction to categorified colored Jones invariants and invariants of 3-manifolds will be studied in detail in subsequent papers.

Categorification of the Jones–Wenzl projectors

“…We would like to mention that, while preparing this manuscript for publication, during the MSRI workshop "Homology Theories of Knots and Links" in March 2010 we learned that an alternative, representation-theoretic, approach to categorifying the Jones-Wenzl projectors has been pursued by Igor Frenkel, Catharina Stroppel and Joshua Sussan [9]. In light of the universality properties of our construction (see Section 3), it is plausible that the two approaches are equivalent, although the methods are quite different.…”

“…One advantage of working in Khovanov's and BarNatan's framework for categorification of the Temperley-Lieb algebra is that our construction of the categorified projectors is explicit and it is readily available for topological applications. The interested reader may want to compare our construction in Section 6.2 to the discussion of the 6j -symbols in [9], Section 17.…”

Super-A-polynomials for twist knots

“…After the pioneering works by Khovanov [10,11], the bi-graded homology theory called the colored sl 2 knot homology H sl 2 ,R i,j was introduced [12][13][14] as a categorification of the colored Jones polynomial. We denote the Poincaré polynomial of the colored sl 2 homology H sl 2 ,R i,j by P sl 2 R (K; q, t) = i,j 4) so that the subscripts i and j are called the quantum (polynomial) grading and the homological gradings respectively.…”

“…More precisely, it is conjectured in [52] that, taking double scaling limit, the equation 14) gives the zero locus of the A-polynomial A(K; x, y) of the knot K. Physically, it is natural to quantize the A-polynomial A(K; x, y), which result in the operator A(K;x,ŷ; q). Taking q = e = 1 gives the classical A-polynomial A(K; x, y).…”

Lectures on Knot Homology and Quantum Curves

“…Much like the Khovanov homology of a knot is a categorification of its Jones polynomial or quantum sl(2) invariant, there exist generalizations [57,56,48,8,20] of the Khovanov homology categorifying the n-colored Jones polynomials for all n:…”