4-manifold days in Brisbane

schedule

All talks will be held in Carslaw building, room 535A.
Morning and afternoon tea will be served in the tea room on level 7 in Carslaw building (room 727).
Lunch will also be provided in Carslaw 727 on Thursday and Friday.

Wed 2 April Thu 3 April Fri 4 April

09:00-09:50 Stefan Friedl Mark Bell

10:00-10:50 Joachim Gudmundsson Jonathan Spreer

10:50-11:30 Morning tea Morning tea

11:30-12:20 Joan Licata Hyam Rubinstein

12:20-14:20 Lunch Lunch Lunch

14:20-15:10 Saul Schleimer Ben Burton

15:10-16:00 Afternoon tea Afternoon tea
16:00-16:50 Nathan Dunfield Jessica Purcell

19:00 Conference dinner

talks

Mark Bell (University of Warwick)

TITLE Deciding Nielsen-Thurston types of surface diffeomorphisms

ABSTRACT Given a surface diffeomorphism as a word in some fixed generating set, one approach to the problem of deciding its Nielsen-Thurston type is to decide if it fixes a curve (up to isotopy). We will discuss an algorithm to do this which runs in polynomial time on a non-deterministic Turing machine. One consequence of this is that for each diffeomorphism there is a proof of its NT-type whose length is a most a polynomial in the length of the corresponding word.

Ben Burton (The University of Queensland)

TITLE Exploring parameterised complexity in computational topology

ABSTRACT Decision problems on 3-manifolds are notoriously difficult: the mere existence of an algorithm is often a major result, even "simple" problems have best-known algorithms that are exponential time, and many important algorithms have yet to be studied from a complexity viewpoint. In practice, however, some of these algorithms run surprisingly well in experimental settings. In this talk we discuss why parameterised complexity now looks to be the "right" tool to explain this behaviour. We present some initial forays into the parameterised complexity of topological problems, including both fixed-parameter-tractability and W[P]-hardness results. Includes joint work with Rod Downey, Thomas Lewiner, João Paixão, William Pettersson, and Jonathan Spreer.

Nathan Dunfield (University of Illinois at Urbana-Champaign)

TITLE Floer homology and orderability of 3-manifold groups

ABSTRACT I will discuss the interconnections between Heegaard Floer homology, taut foliations, and actions of 3-manifold groups on 1-dimensional spaces. The focus will be a conjecture of Boyer-Gordon-Watson about rational homology 3-spheres, namely that having nonminimal Heegaard Floer homology is equivalent to the fundamental group acting on the line. I will show experimental evidence that supports this conjecture, namely a partial survey of these properties for the roughly 11,000 manifolds in the Hodgson-Weeks census as well as some 160,000 double-branched covers of links with at most 15 crossings.

Stefan Friedl (University of Regensburg)

TITLE The profinite completion of knot groups

ABSTRACT The profinite completion of a group contains the same information as the set of all finite quotients of a group. We will show that the profinite completion of a knot group determines whether or not the knot is fibered. We furthermore show that the profinite completion detects the trefoil and the Figure-8 knot.

Joachim Gudmundsson (The University of Sydney)

TITLE Geometric spanner graphs

ABSTRACT Consider a set S of n points in d-dimensional Euclidean space. A network on S can be modelled as an undirected graph G with vertex set S of size n and an edge set E, where every edge (u,v) has a weight. A geometric (Euclidean) network is a network where the weight of the edge (u,v) is the Euclidean distance between its endpoints. Given a real number t > 1 we say that G is a t-spanner for S, if for each pair of points u,v in S, there exists a path in G of weight at most t times the Euclidean distance between u and v. In this talk we give an overview of the area of geometric spanners including the most common construction algorithms to obtain spanners of linear size.

Joan Licata (Australian National University)
TITLE Front diagrams via open book decompositions

ABSTRACT Grid diagrams offer a simple combinatorial presentation of a knot in a lens space; in the case that the lens space is equipped with its universally tight contact contact structure, grid diagrams can also be used to present Legendrian knots. This "front projection" has lead to a variety of algorithms and computer programs for computing invariants of Legendrian knots in contact lens spaces. In this talk I'll discuss joint work with Dave Gay to develop an analogous notion of front projection for Legendrian knots in arbitrary contact manifolds.

Jessica Purcell (Brigham Young University)

TITLE Geometrically maximal knots

ABSTRACT In this talk, we consider the ratio of volumes of hyperbolic knots to their crossing numbers. This ratio is known to have maximum value less than the volume of a regular ideal octahedron. This motivates several questions, such as, for which knots is the ratio very near the maximum? For fixed crossing number, what links maximize this ratio? We say that a sequence of hyperbolic knots is geometrically maximal if these ratios limit to the maximum value. In this talk, we describe several sequences of geometrically maximal knots, and present several conjectures. We discuss weaving knots, which are alternating knots with the same projection as a torus knot, and which were conjectured by Lin to be among the maximum volume knots for fixed crossing number. We prove weaving knots are geometrically maximal. We discuss a method, known to Agol, for constructing other sequences of geometrically maximal knots. This is joint with Abhijit Champanerkar and Ilya Kofman.

Hyam Rubinstein (The University of Melbourne)

TITLE Even triangulations of manifolds

ABSTRACT This is joint work with Stephan Tillmann and partly with Marcel Bokstedt (Aarhus). Even triangulations have the property that each codimension two face has even degree. They have remarkable properties – in particular symmetry representations of the fundamental group into finite permutation groups. The symmetry covering has the property that one can consistently colour objects such as faces and vertices of the simplices. In particular, one can only find embedded normal surfaces which entirely consist of quadrilaterals in the 3-dimensional case. We give some general constructions and characterisations of when such triangulations exist. This is related to Heegaard splittings and existence of mod 2 homology. Finally we give a general construction of Cat(0) structures on such triangulations based on additional properties of even triangulations.

Saul Schleimer (University of Warwick)

TITLE "Fibered class" lies in NP

ABSTRACT (Joint work with Ben Burton.) The decision problem "Fibered class" asks, given a triangulated three-manifold M and a one-dimensional cohomology class α, is α Poincaré dual to a fiber of a fibration of M? This is closely related to the problem of determining the Thurston norm of a given class α. We show "Fibered class" lies in NP.

Jonathan Spreer (The University of Queensland)

TITLE Bounds for the genus of a normal surface

ABSTRACT We present a sharp bound on the genus of a normal surface in a triangulated compact, orientable 3-manifold in terms of the quadrilaterals in its cell decomposition. In addition, we describe two applications of this bound: In the first, we use it to determine the minimal triangulations of the product of a closed surface and the closed interval. In the second, we show that an alternative approach of the realisation problem using normal surface theory is less powerful than its dual method using subcomplexes of polytopes. This is joint work with William Jaco, Jesse Johnson, and Stephan Tillmann.

Back to the main page.

http://www.maths.usyd.edu.au/u/tillmann/cats2014/
17 Februar 2014

	Wed 2 April	Thu 3 April	Fri 4 April
09:00-09:50		Stefan Friedl	Mark Bell
10:00-10:50		Joachim Gudmundsson	Jonathan Spreer
10:50-11:30		Morning tea	Morning tea
11:30-12:20		Joan Licata	Hyam Rubinstein
12:20-14:20	Lunch	Lunch	Lunch
14:20-15:10	Saul Schleimer	Ben Burton
15:10-16:00	Afternoon tea	Afternoon tea
16:00-16:50	Nathan Dunfield	Jessica Purcell
19:00	Conference dinner

	Mark Bell (University of Warwick)
TITLE	Deciding Nielsen-Thurston types of surface diffeomorphisms
ABSTRACT	Given a surface diffeomorphism as a word in some fixed generating set, one approach to the problem of deciding its Nielsen-Thurston type is to decide if it fixes a curve (up to isotopy). We will discuss an algorithm to do this which runs in polynomial time on a non-deterministic Turing machine. One consequence of this is that for each diffeomorphism there is a proof of its NT-type whose length is a most a polynomial in the length of the corresponding word.

	Ben Burton (The University of Queensland)
TITLE	Exploring parameterised complexity in computational topology
ABSTRACT	Decision problems on 3-manifolds are notoriously difficult: the mere existence of an algorithm is often a major result, even "simple" problems have best-known algorithms that are exponential time, and many important algorithms have yet to be studied from a complexity viewpoint. In practice, however, some of these algorithms run surprisingly well in experimental settings. In this talk we discuss why parameterised complexity now looks to be the "right" tool to explain this behaviour. We present some initial forays into the parameterised complexity of topological problems, including both fixed-parameter-tractability and W[P]-hardness results. Includes joint work with Rod Downey, Thomas Lewiner, João Paixão, William Pettersson, and Jonathan Spreer.

	Nathan Dunfield (University of Illinois at Urbana-Champaign)
TITLE	Floer homology and orderability of 3-manifold groups
ABSTRACT	I will discuss the interconnections between Heegaard Floer homology, taut foliations, and actions of 3-manifold groups on 1-dimensional spaces. The focus will be a conjecture of Boyer-Gordon-Watson about rational homology 3-spheres, namely that having nonminimal Heegaard Floer homology is equivalent to the fundamental group acting on the line. I will show experimental evidence that supports this conjecture, namely a partial survey of these properties for the roughly 11,000 manifolds in the Hodgson-Weeks census as well as some 160,000 double-branched covers of links with at most 15 crossings.

	Stefan Friedl (University of Regensburg)
TITLE	The profinite completion of knot groups
ABSTRACT	The profinite completion of a group contains the same information as the set of all finite quotients of a group. We will show that the profinite completion of a knot group determines whether or not the knot is fibered. We furthermore show that the profinite completion detects the trefoil and the Figure-8 knot.

	Joachim Gudmundsson (The University of Sydney)
TITLE	Geometric spanner graphs
ABSTRACT	Consider a set S of n points in d-dimensional Euclidean space. A network on S can be modelled as an undirected graph G with vertex set S of size n and an edge set E, where every edge (u,v) has a weight. A geometric (Euclidean) network is a network where the weight of the edge (u,v) is the Euclidean distance between its endpoints. Given a real number t > 1 we say that G is a t-spanner for S, if for each pair of points u,v in S, there exists a path in G of weight at most t times the Euclidean distance between u and v. In this talk we give an overview of the area of geometric spanners including the most common construction algorithms to obtain spanners of linear size.

	Joan Licata (Australian National University)
TITLE	Front diagrams via open book decompositions
ABSTRACT	Grid diagrams offer a simple combinatorial presentation of a knot in a lens space; in the case that the lens space is equipped with its universally tight contact contact structure, grid diagrams can also be used to present Legendrian knots. This "front projection" has lead to a variety of algorithms and computer programs for computing invariants of Legendrian knots in contact lens spaces. In this talk I'll discuss joint work with Dave Gay to develop an analogous notion of front projection for Legendrian knots in arbitrary contact manifolds.

	Jessica Purcell (Brigham Young University)
TITLE	Geometrically maximal knots
ABSTRACT	In this talk, we consider the ratio of volumes of hyperbolic knots to their crossing numbers. This ratio is known to have maximum value less than the volume of a regular ideal octahedron. This motivates several questions, such as, for which knots is the ratio very near the maximum? For fixed crossing number, what links maximize this ratio? We say that a sequence of hyperbolic knots is geometrically maximal if these ratios limit to the maximum value. In this talk, we describe several sequences of geometrically maximal knots, and present several conjectures. We discuss weaving knots, which are alternating knots with the same projection as a torus knot, and which were conjectured by Lin to be among the maximum volume knots for fixed crossing number. We prove weaving knots are geometrically maximal. We discuss a method, known to Agol, for constructing other sequences of geometrically maximal knots. This is joint with Abhijit Champanerkar and Ilya Kofman.

	Hyam Rubinstein (The University of Melbourne)
TITLE	Even triangulations of manifolds
ABSTRACT	This is joint work with Stephan Tillmann and partly with Marcel Bokstedt (Aarhus). Even triangulations have the property that each codimension two face has even degree. They have remarkable properties – in particular symmetry representations of the fundamental group into finite permutation groups. The symmetry covering has the property that one can consistently colour objects such as faces and vertices of the simplices. In particular, one can only find embedded normal surfaces which entirely consist of quadrilaterals in the 3-dimensional case. We give some general constructions and characterisations of when such triangulations exist. This is related to Heegaard splittings and existence of mod 2 homology. Finally we give a general construction of Cat(0) structures on such triangulations based on additional properties of even triangulations.

	Saul Schleimer (University of Warwick)
TITLE	"Fibered class" lies in NP
ABSTRACT	(Joint work with Ben Burton.) The decision problem "Fibered class" asks, given a triangulated three-manifold M and a one-dimensional cohomology class α, is α Poincaré dual to a fiber of a fibration of M? This is closely related to the problem of determining the Thurston norm of a given class α. We show "Fibered class" lies in NP.

	Jonathan Spreer (The University of Queensland)
TITLE	Bounds for the genus of a normal surface
ABSTRACT	We present a sharp bound on the genus of a normal surface in a triangulated compact, orientable 3-manifold in terms of the quadrilaterals in its cell decomposition. In addition, we describe two applications of this bound: In the first, we use it to determine the minimal triangulations of the product of a closed surface and the closed interval. In the second, we show that an alternative approach of the realisation problem using normal surface theory is less powerful than its dual method using subcomplexes of polytopes. This is joint work with William Jaco, Jesse Johnson, and Stephan Tillmann.