## Schedule

All talks will be held in CBE Lecture Theatre 4. (This is in the building immediately north-east of John Dedman.)
Registration and tea breaks will be held in the Allen Barton Forum (also in the CBE building).

### Monday introductory talks

8:30-9:15
Tea and registration
9:15-9:30
Welcome
9:30-10:30
Introduction to 3-manifolds - Stephan Tillmann
11:00-12:00
Introduction to link homologies - Hoel Queffelec
1:30-2:30
Introduction to finite type invariants - Zsuzsi Dancso
3:00-4:00
Introduction to contact geometry - Daniel Matthews

### Research talks Tuesday-Friday

 Tuesday Wednesday Thursday Friday 9:00-10:00 Craig Baptiste Jessica Anthony 10:10-11:10 Hyam Joan Norm Hoel 11:40-12:40 Jonathan Dan Gwenael Scott 2:00-3:00 Stephan Josh Radmila --- 3:30-4:30 Ben Yoav Zsuzsi ---

A list of all participants, if you're as bad at remembering names as I am, is available here.

## Program Description

The MSI Workshop on Low-Dimensional Topology & Quantum Algebra will be held October 31-November 4 (Monday-Friday) at the Australian National University. The workshop is part of the MSI's Special Year program on Algebra and Topology.
Although in the past they have been viewed as separate fields, low-dimensional topology and quantum algebra have recently seen surprising interactions with the development of new topological invariants with deep connections to quantum groups and category theory. These invariants have proven to be effective tools for tackling fundamental problems in manifold and link theory, but they also have generated new research activity in algebra due to their compelling internal structure with close connections to higher categories and higher representation theory. This workshop will focus on recent advances in this developing subject, bringing together researchers from all parts of this vibrant field, and highlighting work of both established experts and exceptional young researchers from inside and outside of Australia.
The conference poster is available.
Registration is now open. Please register to let us know you're coming!

## Introductory talks

Why be happy when you could be normal? (Stephan Tillmann, Sydney)
Computation, experimentation and conjectures have played a driving role in low-dimensional topology right from the cradle of modern topology in the work of Poincare. There is a successful history of replacing non-constructive existence proofs by practical solutions, and heuristic methods by rigorous algorithms. In this introductory talk, I will describe some of the key techniques used to study a 3-dimensional space, including finding essential surfaces in the space, constructing a geometric structure on it, and computing key invariants. If time permits, I will also outline some open problems on 3-manifolds and related fields.
Finite type invariants and the Kontsevich Integral (Zsuzsanna Dancso, ANU)
This talk is an introduction to the Vassiliev filtration on knots, chord diagram spaces, finite type invariants and universal finite type invariants, and the universal finite type invariant of knots called the Kontsevich integral.
Contact geometry (Daniel Matthews, Monash)

## Abstracts

TBA (Benjamin Burton, Queensland)
Positive Legendrian isotopies and Floer theory (Baptiste Chantraine, Nantes)
In a cooriented contact manifold, a positive Legendrian isotopy is a Legendrian isotopy evolving in the positive transverse direction to the contact plane. Their global behavior differs from the one of Legendrian isotopy and is closer to the one of propagating waves. In this talk I will explain how to use information in the Floer complex associated to a pair of Lagrangian cobordisms (recently constructed in a collaboration with G. Dimitroglou Rizell, P. Ghiggini and R. Golovko) to give obstructions to certain positive loops of some Legendrian submanifolds. This will recover previously known obstructions and exhibit more examples. This is work in progress with V. Colin and G. Dimitroglou Rizell.
Topological recursion and quantum curves (Norm Do, Monash)
Topological recursion is a machinery that arose in physics but has recently found widespread application to diverse areas of mathematics. Whenever a problem is governed by the topological recursion, there is usually a quantum curve lurking about as well. We will demonstrate this circle of ideas using a toy example from very low-dimensional topology before discussing connections with the volume and AJ conjectures from quantum topology.
Finite type invariants of welded knots and the Alexander polynomial (Zsuzsanna Dancso, ANU)
This talk is based on joint work with Dror Bar-Natan. I will explain the parallels and differences between the theories of finite type invariants for classical and welded knotted objects. In particular, a version of the Alexander polynomial is a universal finite type invariant for welded knots.
Counting genus 2 surfaces in 3-manifolds (Craig Hodgson, Melbourne)
The 3d-index of a hyperbolic knot complement $M$ is a powerful invariant introduced by the physicists Dimofte, Gaiotto and Gukov, giving a collection of formal power series in q with integer coefficients, indexed by a pair of integers.
When both integers are zero, the constant term is 1 and we show that the coefficient of $q$ can be expressed in terms of the numbers of genus 2 normal and almost normal surfaces in a suitable ideal triangulation of $M$. Further, this coefficient also has a purely topological description, in terms of the numbers of isotopy classes of genus 2 Heegaard surfaces and genus 2 incompressible surfaces in $M$. We will give examples illustrating these results, and sketch an approach to the proof, which also gives algorithms for counting isotopy classes of the above genus 2 surfaces.
(This is based on joint work with Stavros Garoufalidis, Hyam Rubinstein, Henry Segerman and Neil Hoffman.)
Uniformly Twisted Knots and the Slope Conjectures (Josh Howie, Melbourne)
The slope conjectures of Garoufalidis and Kalfagianni-Tran predict that the highest and lowest degrees of the coloured Jones polynomial contain topological information about essential surfaces properly embedded in the knot exterior. Uniformly twisted knots were introduced recently by Ozawa, and these knots contain a pair of essential surfaces in the knot exterior; one a spanning surface, and the other a coiled surface. We show that these two surfaces coincide with the two Jones surfaces for many semi-adequate knots with integral Jones slopes. This allows us to complete the proof that all knots with up to 9 crossings satisfy the strong slope conjecture.
Morse Structures on Open Books (Joan Licata, ANU)
Every contact 3-manifold is locally contactomorphic to the standard contact $\mathbb R^3$, but this fact does not necessarily produce large charts that cover the manifold efficiently. I'll describe joint work with Dave Gay which uses an open book decomposition of a contact manifold to produce a particularly efficient collection of such contactomorphisms, together with simple combinatorial data describing how to reconstruct the contact 3-manifold from these charts. We use this construction to define front projections for Legendrian knots and links in arbitrary contact 3-manifolds, generalising existing constructions of front projections for Legendrian knots in $S^3$ and universally tight lens spaces.
The braid group and 2-representation theory (Tony Licata, ANU)
The symmetric group $S_n$ has a faithful $(n-1)$-dimensional representation, and much of the rich combinatorics of permutations (e.g. the study of reduced expressions and Bruhat order) can be understood via linear algebra using this representation. One can study the braid group $B_n$ somewhat analogously, at the cost of embracing one more level of categorical abstraction. The goal of this talk will be to explain (to topologists!) what one learns about the braid group $B_n$ by studying a faithful action of $B_n$ on a triangulated category.
(This is joint work with Hoel Queffelec.)
Strand Algebras and Contact Categories (Dan Mathews, Monash)
In the Bordered Heegaard Floer homology theory of Lipshitz-Oszvath-Szabo, an algebra known as a "strand algebra" is associated to a surface. Zarev defined a generalisation of this algebra for Bordered Sutured Heegaard Floer homology. These algebras, while describing the behaviour of holomorphic discs near the boundary of a 3-manifold, can be defined purely combinatoriallly as a differential graded algebra of combinatorial "strand diagrams". In recent work, we have shown that this algebra can be given an elementary description in terms of contact geometry. In particular, its homology is the algebra of a contact category. In this talk we will describe some of the ideas involved.
Extended Kontsevich integral for bottom tangles in handlebodies. (Gwenael Massuyeau, Strasbourg)
(This is joint work with Kazuo Habiro.)
Using the Kontsevich integral, we construct a functor $Z$ from the category $B$ of "bottom tangles in handlebodies" to a certain category $\mathcal A$ of "Jacobi diagrams in handlebodies". Thus we show that the completion of $B$ with respect to the Vassiliev filtration is isomorphic to $\mathcal A$, and we give a presentation of the latter as a symmetric monoidal category. If time allows, we will also explain how $Z$ relates to the LMO functor, which is a kind of TQFT derived from the Le-Murakami-Ohtsuki invariant of homology 3-spheres.
The curve complex and 3-manifolds (Yoaz Moriah, Technion)
Expository talk about the curve complex and its relevance to the study of 3-manifolds.
LYMPH TOFU, Dubrovnik, and the Quantum Exceptional Series (Scott Morrison, ANU)
Exceptional Dehn fillings and the geometry of planar surfaces. (Jessica Purcell, Monash)
When a 3-manifold with torus boundary is Dehn filled, and the result is not hyperbolic, the Dehn filling is called exceptional. Exceptional Dehn fillings have been studied topologically and geometrically for many years. The 6-theorem implies that if a Dehn filling is exceptional, then the length of the slope of the filling is at most 6. Agol showed that 6 is a sharp bound for toroidal fillings. However, there are several other types of exceptional Dehn fillings, and the optimal bounds on the lengths of corresponding slopes are unknown. In this talk, we will construct hyperbolic 3-manifolds with reducible Dehn fillings with the longest known slopes, and discuss lengths of other exceptional Dehn fillings. We give conjectured and experimental bounds on their slope lengths. This is joint work with Neil Hoffman.
Around Chebyshev's polynomial and the skein algebra of the torus (Hoel Queffelec, Montpellier)
(Joint work with H. Russell, D. Rose and P. Wedrich)
The diagrammatic version of the Jones polynomial, based on the Kauffman bracket skein module, extends to knots in any 3-manifold. In the case of thickened surfaces, it can be endowed with the structure of an algebra by stacking. The case of the torus is of particular interest, and C. Frohman and R. Gelca exhibited in 1998 a basis of the skein module for which the multiplication is governed by the particularly simple "product-to-sum" formula.
I'll present a diagrammatic proof of this formula that highlights the role of the Chebyshev's polynomials, before turning to categorification perspectives and their interactions with representation theory.
Some of the topology behind the 3d-index (Hyam Rubinstein, Melbourne)
This is joint work with Nathan Dunfield, Stavros Garoufalidis, Craig Hodgson, Neil Hoffman, and Henry Segerman.
The 3d index is an amazing set of $q$-series associated to a 1-efficient ideal triangulation of a cusped 3-manifold. Since it is a topological invariant, a key problem is to understand what topological information is contained in the coefficients. Currently we are writing up a proof that the coefficient of the linear term of the simplest version of the 3d index counts the number of isotopy classes of incompressible surfaces of genus 2 minus the number of isotopy classes of Heegaard splittings of genus 2.
I will attempt to explain several key technical issues in the proof. The first is that normal and almost normal representatives of incompressible surfaces and Heegaard splittings in isotopy classes can be organised into graphs. For the special case of genus 2, in the presence of a strict angle structure on the triangulation, each such graph is a tree and can be viewed as a Morse complex for the isotopy class. The challenge is to extend this approach to all genera, which involves multi-parameter sweepout theory. We have some ideas about the quadratic term of the index, involving isotopy classes of genus 2 and 3 surfaces of various types and sweepouts up to 4 parameters. In particular, the number of isotopy classes of genus 3 splittings should be part of this count.
Chromatic homology, Khovanov homology, and torsion (Radmila Sazdanovic, North Carolina State)
Experimental computations show that the Khovanov homology of a link tends to have an abundance of torsion. However, torsion of order two appears more frequently than torsion of other orders. We give a partial explanation of this observation, at least in the first and/or last few homological gradings of Khovanov homology. There is a partial isomorphism between the Khovanov homology of a link and the chromatic polynomial categorification of a certain graph related to a diagram of the link. We show that the chromatic polynomial categorification contains only torsion of order two, and consequently, Khovanov homology can only contain torsion of order two in the gradings where the partial isomorphism is defined. This is joint work with Adam Lawrence.
The Pachner graph of the 2-sphere (Jonathan Spreer, Queensland)
It is well-known that any two $n$-vertex triangulations of the 2-sphere are connected by a sequence of edge flips. In other words, the Pachner graph of $n$-vertex 2-sphere triangulations is connected. In this article, we study various induced subgraphs of this graph. In particular, we prove that the subgraph induced by the set of $n$-vertex flag 2-spheres distinct from the double cone is still connected. In contrast, we show that the subgraph induced by the $n$-vertex stacked spheres has at least as many components as there are cubic trees on $n/3$ vertices.
Multisections of manifolds (Stephan Tillmann, Sydney)
Dave Gay and Rob Kirby recently introduced trisections of smooth 4-manifolds arising from their study of broken Lefschetz fibrations and Morse 2-functions. Dave asked us if this could be established using triangulations. We have done this and extended the theory to all dimensions. The idea is to split a $2k$- or $(2k+1)$-manifold into $k$ handlebodies, such that intersections of the handlebodies have special properties. The splitting can be viewed as mapping the manifold into a $k$-simplex and pulling back a decomposition into dual cubes. I'll outline the construction, give some applications and conclude with open questions. This is joint work with Hyam Rubinstein.

## Financial support

This event is sponsored by the Australian Mathematical Sciences Institute (AMSI) and the Australian Mathematical Society. AMSI allocates a travel allowance annually to each of its member universities. Students or early career researchers from AMSI member universities without access to a suitable research grant or other source of funding may apply to the Head of Mathematical Sciences for subsidy of travel and accommodation out of the departmental travel allowance.