We develop a rigorous algebraic background for representations of triangle groups in linear groups over algebras arising from factor rings of multivariate polynomial rings. This is then used to substantially improve the existing upper bounds on the order of epimorphic images of triangle groups with a given injectivity radius and, analogously, the size of the associated hypermaps of a given type...

Abstract. The volume conjecture and its generalization state that the series of certain evaluations of the colored Jones polynomials of a knot would grow exponentially and its growth rate would be related to the volume of a threemanifold obtained by Dehn surgery along the knot. In this paper, we show that for the figure-eight knot the series converges in some cases and the limit equals the inve...

In this paper we show that the polynomial x − 5x +12x − 28x +72x − 132x + 116x − 74x + 90x − 28x − 12x + 24x − 12x − 4x − 3x − 1 ∈ Q[x] has Galois group SL2(F16), filling in a gap in the tables of Jürgen Klüners and Gunther Malle (see [12]). The computation of this polynomial uses modular forms and their Galois representations.

Dowling lattices and their generalizations introduced by Hanlon are interpreted as lattices of congruences associated to certain quasi-varieties of sets with group actions. This interpretation leads, by a simple application of Mobius inversion, to polynomial identities which specialize to Hanlon's evaluation of the characteristic polynomials of generalized Dowling lattices. Analogous results ar...

This work explores a method that reduces the design of evaluation strategies for geometric predicates to the computation of polynomial invariants of a group action. We apply it to the classical problem of counting line transversals to lines in P and capture polynomials previously obtained by more pedestrian approaches.

We consider a cyclic group of order p acting on a module incharacteristic p and show how to reduce the calculation of the symmetric algebra to that of the exterior algebra. Consider a cyclic group of order p acting on a polynomial ring S = k[x1, . . . , xr], where k is a field of characteristic p; this is equivalent to the symmetric algebra S∗(V ) on the module V generated by x1, . . . , xr. We...

The Poincaré polynomial of a Weyl group calculates the Betti numbers of the projective homogeneous space G/B, while the h-vector of a simple polytope calculates the Betti numbers of the corresponding rationally smooth toric variety. There is a common generalization of these two extremes called the H-polynomial. It applies to projective, homogeneous spaces, toric varieties and, much more general...

We construct link invariants using the D2n subfactor planar algebras, and use these to prove new identities relating certain specializations of colored Jones polynomials to specializations of other quantum knot polynomials. These identities can also be explained by coincidences between small modular categories involving the even parts of the D2n planar algebras. We discuss the origins of these ...