Abstract. A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or group-like object. It is also known as a spherical category, or a strict, monoidal category with a few extra properties, or by several other names. A recently useful point of view, developed by other authors, of the representation theory of sl(2) has been to present it as a spider by generators and relations. That is, one has an algebraic spider, defined by invariants of linear representations, and one identifies it as isomorphic to a combinatorial spider, given by generators and relations. We generalize this approach to the rank 2 simple Lie algebras, namely A 2 , B 2 , and G 2 . Our combinatorial rank 2 spiders yield bases for invariant spaces which are probably related to Lusztig's canonical bases, and they are useful for computing quantities such as generalized 6j-symbols and quantum link invariants. Their definition originates in definitions of the rank 2 quantum link invariants that were discovered independently by the author and Francois Jaeger.

In a previous article [23], we derived the alternating-sign matrix (ASM) theorem from the Izergin-Korepin determinant [12,13,19] for a partition function for square ice with domain wall boundary. Here we show that the same argument enumerates three other symmetry classes of alternating-sign matrices: VSASMs (vertically symmetric ASMs), even HTSASMs (half-turn-symmetric ASMs), and even QTSASMs (quarter-turn-symmetric ASMs). The VSASM enumeration was conjectured by Mills; the others by Robbins [31]. We introduce several new types of ASMs: UASMs (ASMs with a U-turn side), UUASMs (two U-turn sides), OSASMs (offdiagonally symmetric ASMs), OOSASMs (off-diagonally, off-antidiagonally symmetric), and UOSASMs (offdiagonally symmetric with U-turn sides). UASMs generalize VSASMs, while UUASMs generalize VHSASMs (vertically and horizontally symmetric ASMs) and another new class, VHPASMs (vertically and horizontally perverse). OSASMs, OOSASMs, and UOSASMs are related to the remaining symmetry classes of ASMs, namely DSASMs (diagonally symmetric), DASASMs (diagonally, anti-diagonally symmetric), and TSASMs (totally symmetric ASMs). We enumerate several of these new classes, and we provide several 2-enumerations and 3-enumerations.Our main technical tool is a set of multi-parameter determinant and Pfaffian formulas generalizing the Izergin-Korepin determinant for ASMs and the Tsuchiya determinant for UASMs [39]. We evaluate specializations of the determinants and Pfaffians using the factor exhaustion method.

We present a quantum algorithm for the dihedral hidden subgroup problem with time and query complexity 2 O( √ log N) . In this problem an oracle computes a function f on the dihedral group D N which is invariant under a hidden reflection in D N . By contrast the classical query complexity of DHSP is O( √ N). The algorithm also applies to the hidden shift problem for an arbitrary finitely generated abelian group.The algorithm begins as usual with a quantum character transform, which in the case of D N is essentially the abelian quantum Fourier transform. This yields the name of a group representation of D N , which is not by itself useful, and a state in the representation, which is a valuable but indecipherable qubit. The algorithm proceeds by repeatedly pairing two unfavorable qubits to make a new qubit in a more favorable representation of D N . Once the algorithm obtains certain target representations, direct measurements reveal the hidden subgroup.

We establish a 3-manifold invariant for each finite-dimensional, involutory Hopf algebra. If the Hopf algebra is the group algebra of a group G, the invariant counts homomorphisms from the fundamental group of the manifold to G. The invariant can be viewed as a state model on a Heegaard diagram or a triangulation of the manifold. The computation of the invariant involves tensor products and contractions of the structure tensors of the algebra. We show that every formal expression involving these tensors corresponds to a unique 3-manifold modulo a well-understood equivalence. This raises the possibility of an algorithm which can determine whether two given 3-manifolds are homeomorphic.

Several authors have recently studied virtual knots and links because they admit invariants arising from R-matrices. We prove that every virtual link is uniquely represented by a link L ⊂ S ×I in a thickened, compact, oriented surface S such that the link complement (S × I) \ L has no essential vertical cylinder. AMS Classification 57M25; 57M27 57M15Keywords Virtual link, tetravalent graph, stable equivalence A virtual link L is an equivalence class of decorated, finite, tetravalent graphs Γ. The edges at each vertex must be cyclically ordered, and two opposite edges are labelled as an overcrossing, while the other two are labelled as an undercrossing. The equivalence relation is the one given by Reidemeister moves. The notion was proposed by Kauffman [5] in light of the fact that R-matrices and quandles, which are commonly used to make link invariants, also yield invariants of virtual links.We will borrow from the analysis of virtual links by Carter, Kamada, and Saito [1]. They show that virtual links are equivalent to stable equivalence classes of links projections onto compact, oriented surfaces. (Fenn, Rourke, and Sanderson have written a self-contained proof [2].) The surface S need not be connected, but we require that each component contains at least one component of the projection P . The projection P is considered up to Reidemeister moves, and the stabilization operation consists of adding a handle to S . Note that the feet of a stabilizing handle may lie in any two regions of the complement of P (Figure 1).As Figure 1 also shows, the reverse destabilization operation consists of cutting the surface S along a circle C which is disjoint from P , and capping the resulting boundary. If C separates the connected component of S of S containing it, then we require that both components of S \ C contain part of P ; otherwise destabilization would create a naked surface component.It is also well-known that a link drawn on a surface S , considered up to Reidemeister moves, is equivalent to a (tame) link L ⊂ S × I in the thickened surface