Scott Edward Morrison

• Ph.D.
• Lecturer at Australian National University

Kashaev and Reshetikhin previously described a way to define holonomy invariants of knots using quantum $\mathfrak{sl}_2$ at a root of unity. These are generalized quantum invariants depend both on a knot $K$ and a representation of the fundamental group of its complement into $\mathrm{SL}_2(\mathbb{C})$; equivalently, we can think of $\mathrm{KR}(...

For a braided fusion category $\mathcal{V}$, a $\mathcal{V}$-fusion category is a fusion category $\mathcal{C}$ equipped with a braided monoidal functor $\mathcal{F}:\mathcal{V} \to Z(\mathcal{C})$. Given a fixed $\mathcal{V}$-fusion category $(\mathcal{C}, \mathcal{F})$ and a fixed $G$-graded extension $\mathcal{C}\subseteq \mathcal{D}$ as an ordi...

For a braided fusion category $\mathcal{V}$, a $\mathcal{V}$-fusion category is a fusion category $\mathcal{C}$ equipped with a braided monoidal functor $\mathcal{F}:\mathcal{V} \to Z(\mathcal{C})$. Given a fixed $\mathcal{V}$-fusion category $(\mathcal{C}, \mathcal{F})$ and a fixed $G$-graded extension $\mathcal{C}\subseteq \mathcal{D}$ as an ordi...

We use Khovanov-Rozansky gl(N) link homology to define pivotal 4-categories, which give rise to invariants of oriented smooth 4-manifolds. The technical heart of this construction is a proof of the sweep-around property, which makes these link homologies well defined in the 3-sphere and implies pivotality for the associated 4-categories.

We establish rank-finiteness for the class of $G$-crossed braided fusion categories, generalizing the recent result for modular categories and including the important case of braided fusion categories. This necessitates a study of slightly degenerate braided fusion categories and their centers, which are interesting for their own sake.

In this paper we construct two new fusion categories and many new subfactors related to the exceptional Extended Haagerup subfactor. The Extended Haagerup subfactor has two even parts EH1 and EH2. These fusion categories are mysterious and are the only known fusion categories which appear to be unrelated to finite groups, quantum groups, or Izumi q...

Monoidal categories enriched in a braided monoidal category $\mathcal{V}$ are classified by braided oplax monoidal functors from $\mathcal{V}$ to the Drinfeld centers of ordinary monoidal categories. In this article, we prove that this classifying functor is strongly monoidal if and only if the original $\mathcal{V}$-monoidal category is tensored o...

A formula for the modular data of $\mathcal{Z}(Vec^{\omega}G)$ was given by Coste, Gannon and Ruelle in arXiv:arch-ive/0001158, but without an explicit proof for arbitrary 3-cocycles. This paper presents a derivation using the representation category of the quasi Hopf algebra $D^{\omega}G$. Further, we have written code to compute this modular data...

We compute the modular data (that is, the $S$ and $T$ matrices) for the centre of the extended Haagerup subfactor. The full structure (i.e. the associativity data, also known as 6-$j$ symbols or $F$ matrices) still appears to be inaccessible. Nevertheless, starting with just the number of simple objects and their dimensions (obtained by a combinato...

We collate information about the fusion categories with $A_n$ fusion rules. This note includes the classification of these categories, a realisation via the Temperley-Lieb categories, the auto-equivalence groups (both braided and tensor), identifications of the subcategories of invertible objects, and explicit descriptions of the Drinfeld centres....

In this note, we discuss the notion of symmetric self-duality of shaded planar algebras, which allows us to lift shadings on subfactor planar algebras to obtain Z/2Z-graded unitary fusion categories. This finishes the proof that there are unitary fusion categories with fusion graphs 4442 and 3333.

This is the first paper in a general program to automate skein theoretic arguments. In this paper, we study skein theoretic invariants of planar trivalent graphs. Equivalently, we classify trivalent categories, which are nondegenerate pivotal tensor categories over generated by a symmetric self-dual simple object X and a rotationally invariant morp...

We introduce the notion of a monoidal category enriched in a braided monoidal category $\mathcal{V}$. We set up the basic theory, and prove a classification result in terms of braided oplax monoidal functors to the Drinfeld centre of some monoidal category $\mathcal{T}$. Even the basic theory is interesting; it shares many characteristics with the...

We introduce the notion of a monoidal category enriched in a braided monoidal category $\mathcal V$. We set up the basic theory, and prove a classification result in terms of braided oplax monoidal functors to the Drinfeld center of some monoidal category $\mathcal T$. Even the basic theory is interesting; it shares many characteristics with the th...

Subfactor standard invariants encode quantum symmetries. The small index subfactor classification program has been a rich source of interesting quantum symmetries. We give the complete classification of subfactor standard invariants to index $5\frac{1}{4}$, which includes $3+\sqrt{5}$, the first interesting composite index.

Subfactor standard invariants encode quantum symmetries. The small index subfactor classification program has been a rich source of interesting quantum symmetries. We give the complete classification of subfactor standard invariants to index $5\frac{1}{4}$, which includes $3+\sqrt{5}$, the first interesting composite index.

We define a third grading on Khovanov homology, which is an invariant of annular links but changes by $\pm 1$ under stabilization. We illustrate the use of our computer implementation, and give some example calculations.

We study unitary quotients of the free product unitary pivotal category A 2 * T 2 . We show that such quotients are parametrized by an integer n ≥ 1 and an 2 n –th root of unity ω . We show that for n = 1, 2, 3, there is exactly one quotient and ω = 1. For 4 ≤ n ≤ 10, we show that there are no such quotients. Our methods also apply to quotients of...

I present a method of calculating the coefficients appearing in the Jones-Wenzl projection in the Temperley-Lieb algebra. It essentially repeats the approach of Frenkel and Khovanov in [4] published in 1997. I wrote this article mid-2002, not knowing about their work, but then set it aside. Recently I decided to dust it off and place it on the arxi...

We introduce a new method for showing that a planar algebra is evaluable. In fact, this method is universal for finite depth subfactor planar algebras. By making careful choices in the method's application, one can often significantly reduce the complexity of the computations. Using our technique, we prove existence and uniqueness of a subfactor pl...

We give the classification of subfactor planar algebras at index exactly 5. All the examples arise as standard invariants of subgroup subfactors. Some of the requisite uniqueness results come from work of Izumi in preparation. The non-existence results build upon the classification of subfactor planar algebras with index less than 5, with some addi...

We explain a technique for discovering the number of simple objects in $Z(C)$, the center of a fusion category $C$, as well as the combinatorial data of the induction and restriction functors at the level of Grothendieck rings. The only input is the fusion ring $K(C)$ and the dimension function $K(C) \to \mathbb{C}$. The method is not guaranteed to...

An irreducible II1-subfactor \({A\subset B}\) is exactly 1-supertransitive if \({B\ominus A}\) is reducible as an A − A bimodule. We classify exactly 1-supertransitive subfactors with index at most \({6\frac{1}{5}}\), leaving aside the composite subfactors at index exactly 6 where there are severe difficulties. Previously, such subfactors were only...

Given a pair of fusion categories $C$ and $D$, we may form the free product $C * D$ and the tensor product $C \boxtimes D$. It is natural to think of the tensor product as a quotient of the free product. What other quotients are possible? When $C=D=A_2$, there is an infinite family of quotients interpolating between the free product and the tensor...

A subfactor is an inclusion $N \subset M$ of von Neumann algebras with trivial centers. The simplest example comes from the fixed points of a group action $M^G \subset M$, and subfactors can be thought of as fixed points of more general group-like algebraic structures. These algebraic structures are closely related to tensor categories and have pla...

We find a new obstruction to the principal graphs of subfactors. It shows that in a certain family of 3-supertransitive principal graphs, there must be a cycle by depth 6, with one exception, the principal graph of the Haagerup subfactor.

One major obstacle in extending the classification of small index subfactors beyond 3+\sqrt{3} is the appearance of infinite families of candidate principal graphs with 4-valent vertices (in particular, the "weeds" Q and Q' from Part 1 (arXiv:1007.1730)). Thus instead of using triple point obstructions to eliminate candidate graphs, we need to deve...

We give a diagrammatic presentation in terms of generators mod relations of the representation category of U_q(sl_n). More precisely, we produce all the relations among SL_n-webs, thus describing the full subcategory tensor-generated by fundamental representations \Alt^k C^n (this subcategory can be idempotent completed to recover the entire repres...

Using Jones' quadratic tangles formulas, we automate the construction of the 4442, 3333, 3311, and 2221 spoke subfactors by finding sets of 1-strand jellyfish generators. The 4442 spoke subfactor is new, and the 3333, 3311, and 2221 spoke subfactors were previously known.

Given an n-manifold M and an n-category C, we define a chain complex (the "blob complex") B *(M;C). The blob complex can be thought of as a derived category analogue of the Hilbert space of a TQFT, and also as a generalization of Hochschild homology to n-categories and n-manifolds. It enjoys a number of nice formal properties, including a higher di...

Progress on classifying small index subfactors has revealed an almost empty landscape. In this paper we give some evidence that this desert continues up to index 3+\sqrt{5}. There are two known quantum-group subfactors with index in this interval, and we show that these subfactors are the only way to realize the corresponding principal graphs. One...

We summarize our axioms for higher categories, and describe the "blob complex." Fixing an n-category , the blob complex associates a chain complex B(*)(W;C) to any n-manifold W. The zeroth homology of this chain complex recovers the usual topological quantum field theory invariants of W. The higher homology groups should be viewed as generalization...

We summarize the known obstructions to subfactors with principal graphs which begin with a triple point. One is based on Jones's quadratic tangles techniques, although we apply it in a novel way. The other two are based on connections techniques; one due to Ocneanu, and the other previously unpublished, although likely known to Haagerup. We then ap...

In this series of papers we show that there are exactly ten subfactors, other than $A_\infty$ subfactors, of index between 4 and 5. Previously this classification was known up to index $3+\sqrt{3}$. In the first paper we give an analogue of Haagerup's initial classification of subfactors of index less than $3+\sqrt{3}$, showing that any subfactor o...

Dimensions of objects in fusion categories are cyclotomic integers, hence number theoretic results have implications in the study of fusion categories and finite depth subfactors. We give two such applications. The first application is determining a complete list of numbers in the interval (2, 76/33) which can occur as the Frobenius-Perron dimensio...

We construct link invariants using the D_2n 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 D_2n...

We give a combinatorial description of the ``$D_{2n}$ planar algebra,'' by generators and relations. We explain how the generator interacts with the Temperley-Lieb braiding. This shows the previously known braiding on the even part extends to a `braiding up to sign' on the entire planar algebra. We give a direct proof that our relations are consist...

Etingof, Nikshych and Ostrik ask in arXiv:math.QA/0203060 if every fusion category can be completely defined over a cyclotomic field. We show that this is not the case: in particular one of the fusion categories coming from the Haagerup subfactor arXiv:math.OA/9803044 and one coming from the newly constructed extended Haagerup subfactor arXiv:0909....

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 gr...

Let V be the 7-dimensional irreducible representation of the quantum group U_q(g_2). For each n, there is a map from the braid group B_n to the endomorphism algebra of the n-th tensor power of V, given by R-matrices. We can extend this linearly to a map on the braid group algebra. Lehrer and Zhang (MR2271576) prove this map is surjective, as a spec...

While topologists have had possession of possible counterexamples to the smooth 4-dimensional Poincar\'{e} conjecture (SPC4) for over 30 years, until recently no invariant has existed which could potentially distinguish these examples from the standard 4-sphere. Rasmussen's s-invariant, a slice obstruction within the general framework of Khovanov h...

We reconsider the su(3) link homology theory defined by Khovanov in math.QA/0304375 and generalized by Mackaay and Vaz in math.GT/0603307. With some slight modifications, we describe the theory as a map from the planar algebra of tangles to a planar algebra of (complexes of) `cobordisms with seams' (actually, a `canopolis'), making it local in the...

This thesis provides a partial answer to a question posed by Greg Kuperberg in q-alg/9712003 and again by Justin Roberts as problem 12.18 in "Problems on invariants of knots and 3-manifolds", math.GT/0406190, essentially: "Can one describe the category of representations of the quantum group U_q(sl_n) (thought of as a spherical category) via genera...

We describe a modification of Khovanov homology (math.QA/9908171), in the spirit of Bar-Natan (math.GT/0410495), which makes the theory properly functorial with respect to link cobordisms. This requires introducing `disorientations' in the category of smoothings and abstract cobordisms between them used in Bar-Natan's definition. Disorientations ha...

We give a simple proof of Lee's result from math.GT/0210213, that the dimension of the Lee variant of the Khovanov homology of an c-component link is 2^c, regardless of the number of crossings. Our method of proof is entirely local and hence we can state a Lee-type theorem for tangles as well as for knots and links. Our main tool is the ``Karoubi e...

We define the pull-back of a smooth principal fibre bundle, and show that it has a natural principal fibre bundle structure. Next, we analyse the relationship between pull-backs by homotopy equivalent maps. The main result of this article is to show that for a principal fibre bundle over a paracompact manifold, there is a principal fibre bundle iso...

This paper gives two applications of Jones's quadratic tangles techniques to the classification of subfactors with index below 5 . In particular, we eliminate two of the five families of possible principal graphs called "weeds" in the classication from part 1. The two families we eliminate here each have principal graph pairs whose first branch poi...

We prove the graph planar algebra embedding theorem, proposed by Vaughan Jones, which states that the representation theory of any finite index II 1 subfactor embeds in the graph planar algebra of its principal graph. In fact, we prove a generalization: for each collection of generators of an arbitrary semisimple pivotal 2-category, we define an as...

Complete definitions and motivations are given for principal fibre bundles, connections on these, and associated vec-tor bundles. The relationships between various concepts of differ-entiation in these settings are proved. Finally, we briefly give an outline of the theory of curvature in terms of connections.