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Stephen Bigelow

Publications and source records attributed to Stephen Bigelow.

At least 19 recordsLinked to original sources
arXiv2026

The Brauer category $\mathcal{B}(2)$ has principal graph $D_\infty$

We show that the subfactor planar algebra with principal graph $D_\infty$ is the Brauer planar algebra with bubble constant $\delta=2$. The Brauer algebra is similar to the Temperley-Lieb algebra, but with virtual crossings. At $\delta=2$, it relates to the category of representations of the orthogonal group $O(2)$. We work over any commutative ring with $1/2$. Its principal graph encodes information about the corresponding monoidal category.

math.RT↗
arXiv2024

Quantum groups from homologies of configuration spaces

We reconstruct a quantum group associated with any Lie algebra together with its representation theory from twisted homologies of generalized configuration spaces of disks. Along the way it brings new combinatorics to the theory, but our diagrams represent true submanifolds of configuration spaces and combinatorial relations between them translate actual twisted homological relations.

math.QA↗
arXiv2018

A new approach to the $SL_n$ spider

The $SL_n$ spider gives a diagrammatic way to encode the representation category of the quantum group $U_q(sl_n)$. The aim of this paper is to define a new spider that contains the $SL_n$ spider. The new spider is defined by generators and relations, according to fairly simple rules that start with combinatorial data coming from the root system of $SL_n$.

math.QA↗
arXiv2018

Realizing an exact entangling gate using Fibonacci anyons

Fibonacci anyons are attractive for use in topological quantum computation because any unitary transformation of their state space can be approximated arbitrarily accurately by braiding. However there is no known braid that entangles two qubits without leaving the space spanned by the two qubits. In other words, there is no known "leakage-free" entangling gate made by braiding. In this paper, we provide a remedy to this problem by supplementing braiding with measurement operations in order to produce an exact controlled rotation gate on two qubits.

quant-ph↗
arXiv2015

The pop-switch planar algebra and the Jones-Wenzl idempotents

The Jones-Wenzl idempotents are elements of the Temperley-Lieb planar algebra that are important, but complicated to write down. We will present a new planar algebra, the pop-switch planar algebra, which contains the Temperley-Lieb planar algebra. It is motivated by Jones' idea of the graph planar algebra of type $A_n$. In the tensor category of idempotents of the pop-switch planar algebra, the $n$th Jones-Wenzl idempotent is isomorphic to a direct sum of $n+1$ diagrams consisting of only vertical strands.

math.QA↗
arXiv2014

Bowling ball representations of braid groups

In a remark in his seminal 1987 paper, Jones describes a way to define the Burau matrix of a positive braid using a metaphor of bowling a ball down a bowling alley with braided lanes. We extend this definition to allow multiple bowling balls to be bowled simultaneously. We obtain the Iwahori-Hecke algebra and a cabled version of the Temperley-Lieb representation.

math.GT↗
arXiv2014

A diagrammatic definition of $U_q(sl_2)$

We give a diagrammatic definition of $U_q(sl_2)$ when $q$ is not a root of unity, including its Hopf algebra structure and its relationship with the Temperley-Lieb category.

math.GT↗
arXiv2014

Alexander representation of tangles

A tangle is an oriented 1-submanifold of the cylinder whose endpoints lie on the two disks in the boundary of the cylinder. Using an algebraic tool developed by Lescop, we extend the Burau representation of braids to a functor from the category of oriented tangles to the category of Z[t,t^{-1}]-modules. For (1,1)-tangles (i.e., tangles with one endpoint on each disk) this invariant coincides with the Alexander polynomial of the link obtained by taking the closure of the tangle. We use the notion of plat position of a tangle to give a constructive proof of invariance in this case.

math.GT↗
arXiv2012

Principal graph stability and the jellyfish algorithm

We show that if the principal graph of a subfactor planar algebra of modulus δ>2 is stable for two depths, then it must end in A_{finite} tails. This result is analogous to Popa's theorem on principal graph stability. We use these theorems to show that an (n-1) supertransitive subfactor planar algebra has jellyfish generators at depth n if and only if its principal graph is a spoke graph.

math.OA↗
arXiv2011

The Alexander and Jones Polynomials Through Representations of Rook Algebras

In the 1920's Artin defined the braid group in an attempt to understand knots in a more algebraic setting. A braid is a certain arrangement of strings in three-dimensional space. It is a celebrated theorem of Alexander that every knot is obtainable from a braid by identifying the endpoints of each string. Because of this correspondence, the Jones and Alexander polynomials, two of the most important knot invariants, can be described completely using the braid group. There has been a recent growth of interest in other diagrammatic algebras, whose elements have a similar topological flavor to the braid group. These have wide ranging applications in areas including representation theory and quantum computation. We consider representations of the braid group when passed through another diagrammatic algebra, the planar rook algebra. By studying traces of these matrices, we recover both the Jones and Alexander polynomials.

math.GT↗
arXiv2011

Constructing the extended Haagerup planar algebra

We construct a new subfactor planar algebra, and as a corollary a new subfactor, with the `extended Haagerup' principal graph pair. This completes the classification of irreducible amenable subfactors with index in the range $(4,3+\sqrt{3})$, which was initiated by Haagerup in 1993. We prove that the subfactor planar algebra with these principal graphs is unique. We give a skein theoretic description, and a description as a subalgebra generated by a certain element in the graph planar algebra of its principal graph. In the skein theoretic description there is an explicit algorithm for evaluating closed diagrams. This evaluation algorithm is unusual because intermediate steps may increase the number of generators in a diagram.

math.OA↗
arXiv2009

Skein theory for the ADE planar algebras

We give generators and relations for the planar algebras corresponding to $ADE$ subfactors. We also give a basis and an algorithm to express an arbitrary diagram as a linear combination of these basis diagrams.

math.QA↗
arXiv2008

Generalized Long-Moody representations of braid groups

Long and Moody gave a method of constructing representations of the braid group B_n. We discuss some ways to generalize their construction. One of these gives representations of subgroups of B_n, including the Gassner representation of the pure braid group as a special case. Another gives representations of the Hecke algebra.

math.GT↗
arXiv2006

A homological definition of the HOMFLY polynomial

We give a new definition of the knot invariant associated to the Lie algebra su_{N+1}. The knot or link must be presented as the plat closure of a braid. The invariant is then a homological intersection pairing between two submanifolds of a configuration space of points in a disk. This generalizes previous work on the Jones polynomial, which is the case N=1.

math.GT↗
arXiv2005

Braid groups and Iwahori-Hecke algebras

This article was submitted to a volume under preparation, with Benson Farb as the editor, on the topic of open problems in surface mapping class groups. The braid group B_n is the mapping class group of an n-times punctured disk. The Iwahori-Hecke algebra H_n is a quotient of the braid group algebra of B_n by a quadratic relation in the standard generators. We discuss how to use H_n to define the Jones polynomial of a knot or link. We also summarize the classification of the irreducible representations of H_n. We conclude with some directions for future research that would apply mapping class group techniques to questions related to H_n.

math.GT↗
arXiv2004

Homological representations of the Iwahori-Hecke algebra

Representations of the Iwahori-Hecke algebra of type A_{n-1} are equivalent to representations of the braid group B_n for which the generators satisfy a certain quadratic relation. We show how to construct such representations from the natural action of B_n on the homology of configuration spaces of the punctured disk. We conjecture that all irreducible representations of Hecke_n can be obtained in this way, even for non-generic values of q.

math.QA↗
arXiv2003

Representations of braid groups

In this paper we survey some work on representations of $B_n$ given by the induced action on a homology module of some space. One of these, called the Lawrence-Krammer representation, recently came to prominence when it was shown to be faithful for all $n$. We will outline the methods used, applying them to a closely related representation for which the proof is slightly easier. The main tool is the Blanchfield pairing, a sesquilinear pairing between elements of relative homology. We discuss two other applications of the Blanchfield pairing, namely a proof that the Burau representation is not faithful for large $n$, and a homological definition of the Jones polynomial. Finally, we discuss possible applications to the representation theory of the Hecke algebra, and ultimately of the symmetric group over fields of non-zero characteristic.

math.GT↗