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Timetable (HTLW02)

3-manifold workshop

Monday 30th January 2017 to Friday 3rd February 2017

Monday 30th January 2017
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from Christie Marr (INI Deputy Director)
10:00 to 11:00 Stefan Friedl (Universität Regensburg); (Universität Regensburg)
Polytope invariants of groups and manifolds
Co-authors: Wolfgang L\"uck (University of Bonn), Kevin Schreve (University of Michigan), Stephan Tillmann (University of Sydney)

We will associate to L^2-acyclic groups and manifolds, modulo some not overly restrictive technical hypothesis, a formal difference of polytopes. We will relate it to the Thurston norm and compute it if the fundamental group has a 2-generator 1-relator presentation
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Marc Lackenby (University of Oxford)
The complexity of unknot recognition
INI 1
12:30 to 13:30 Lunch @ Wolfson Court
14:00 to 15:00 Nathan Dunfield (University of Illinois at Urbana-Champaign)
Floer homology, group orders, and taut foliations of hyperbolic 3-manifolds

A bold conjecture of Boyer-Gorden-Watson and others posit that for any irreducible rational homology 3-sphere M the following three conditions are equivalent: (1) the fundamental group of M is left-orderable, (2) M has non-minimal Heegaard Floer homology, and (3) M admits a co-orientable taut foliation. Very recently, this conjecture was established for all graph manifolds by the combined work of Boyer-Clay and Hanselman-Rasmussen-Rasmussen-Watson. I will discuss a computational survey of these properties involving half a million hyperbolic 3-manifolds, including new or at least improved techniques for computing each of these properties. 

INI 1
15:00 to 15:30 Afternoon Tea
15:30 to 16:30 Sarah Rasmussen (University of Cambridge)
L-space surgeries on iterated satellites by torus links

Satellites by torus links provide multi-component versions of cables.  I will describe the region of L-space surgeries on any torus-link satellite of any L-space knot, with a result that precisely extends Hedden’s and Hom’s analogous result for cables.  More generally, I will characterize the region of L-space surgeries for iterated torus-link satellites, including the case of algebraic links.  This is joint work with Maciej Borodzik.

INI 1
16:30 to 17:30 Welcome Wine Reception at INI
Tuesday 31st January 2017
09:00 to 10:00 Ana Lecuona (Aix Marseille Université)
Slopes, colored links and Kojima's eta concordance invariant
In this talk we will introduce an invariant, the slope, for a colored link in a homology sphere together with a suitable multiplicative character defined on the link group. The slope takes values in the complex numbers union infinity and it is real for finite order characters. It is a generalization of Kojima's eta-invariant and can be expressed as a quotient of Conway polynomials. It is also related to the correction term in Wall’s non-additivity formula for the signatures of 4-manifolds, and as such it appears naturally as a correction term in the expression of the signature formula for the splice of two colored links. This is a work in progress with Alex Degtyarev and Vincent Florens.
INI 1
10:00 to 11:00 Yoav Moriah (Technion - Israel Institute of Technology); (Technion - Israel Institute of Technology)
Diagram Uniqueness for Highly Twisted Plats

 

Co-author: Jessica Purcell (Monash U. Melbourne Australia)
In this paper we prove that if a knot or link has a sufficiently complicated plat projection, then that plat projection is unique. More precisely, if a knot or link has a 2m-plat projection, where m is at least 3, each twist region of the plat contains at least three crossings, and n, the length of the plat, satisfies n > 4m(m − 2), then such a projection is unique up to obvious rotations. In particular, this projection gives a canonical form for such knots and links, and thus provides a classification of these links.

 

INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Masakazu Teragaito (Hiroshima University)
Generalized torsion elements and bi-orderability of 3-manifold groups

 

Co-author: Kimihiko Motegi (Nihon University)  
It is known that a bi-orderable group has no generalized torsion element, but the converse does not hold in general. We conjecture that the converse holds for the fundamental groups of 3-manifolds, and verify the conjecture for non-hyperbolic, geometric 3-manifolds. We also confirm the conjecture for some infinite families of closed hyperbolic 3-manifolds. In the course of the proof, we prove that each standard generator of the Fibonacci group F(2, m) (m > 2) is a generalized torsion element.   

 

INI 1
12:30 to 13:30 Lunch @ Wolfson Court
14:00 to 15:00 Andras Juhasz (University of Oxford)
Cobordism maps in knot Floer homology
Decorated knot cobordisms functorially induce maps on knot Floer homology.
We compute these maps for elementary cobordisms, and hence give a formula for 
the Alexander and Maslov grading shifts. We also show a non-vanishing result in the case of
concordances and present some applications to invertible concordances. 
This is joint work with Marco Marengon.
INI 1
15:00 to 15:30 Afternoon Tea
15:30 to 16:30 Scott Morrison (Australian National University)
The Temperley-Lieb category in operator algebras and in link homology
The Temperley-Lieb category appears in a fundamental way in both the study of subfactors and in link homology theories. Indeed, the discovery of the importance of the Temperley-Lieb category for subfactors led to the creation of the Jones polynomial, and thence, after a long gestation, Khovanov homology.
INI 1
Wednesday 1st February 2017
09:00 to 10:00 Yi Ni (CALTECH (California Institute of Technology))
Null surgery on knots in L-spaces
Co-author: Faramarz Vafaee (Caltech)
 
Let   be a knot in a rational homology sphere  . Then there is a unique surgery on   which results a manifold with    . We call this surgery the null surgery. When   is an L-space, the null surgery remembers the information about the genus and fiberedness of the knot. A special case of our theorem is that if the resulting manifold is      , then the dual knot is a spherical braid. This is joint work with Faramarz Vafaee.
INI 1
10:00 to 11:00 Ken Baker (University of Miami)
Constructions of asymmetric L-space knots

Until July 2014, all known L-spaces admitted an involution.  Then, through a clever search of the SnapPy census, Dunfield-Hoffman-Licata found examples of asymmetric one-cusped hyperbolic manifolds with two lens space fillings and consequently many asymmetric L-space fillings. Yet since none of these lens space fillings were $S^3$, so still stood the conjecture that L-space knots in $S^3$ are strongly invertible.

In this talk we present 
(1) a `natural' realization and vast generalization of the Dunfield-Hoffman-Licata examples (joint work with Hoffman and Licata) and
(2) the first construction of asymmetric L-space knots in $S^3$ (joint work with Luecke).

Both of these constructions produce asymmetric one-cusped hyperbolic manifolds with two fillings that are double branched covers of alternating links, though the approaches are different.

INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 John Luecke (University of Texas at Austin); (University of Texas at Austin)
Boundary-reducing surgeries and bridge number
Let M be a 3–dimensional handlebody of genus g > 1. We give examples of hyperbolic knots in M with arbitrarily large genus g bridge number which admit Dehn surgeries which are boundary- reducible manifolds.
INI 1
12:30 to 13:30 Lunch @ Wolfson Court
14:00 to 15:00 Informal discussion INI 1
15:00 to 15:30 Afternoon Tea
15:30 to 16:30 Informal discussion INI 1
Thursday 2nd February 2017
09:00 to 10:00 Tali Pinsky (Tata Institute of Fundamental Research)
On tunnel number one knots with lens space surgeries
INI 1
10:00 to 11:00 Steven Boyer (UQAM - Université du Québec à Montréal)
Branched covers of quasipositive links and L-spaces

 

Co-authors: Michel Boileau (Université Aix-Marseille), Cameron McA. Gordon (University of Texas at Austin)  
We show that if L is an oriented non-trivial strongly quasipositive link or an oriented quasipositive link which does not bound a smooth planar surface in the 4-ball, then the Alexander polynomial and signature function of L determine an integer n(L) such that \Sigma_n(L), the n-fold cyclic cover of S^3 branched over L, is not an L-space for n > n(L). If K is a strongly quasipositive knot with monic Alexander polynomial such as an L-space knot, we show that \Sigma_n(K) is not an L-space for n \geq 6 and that the Alexander polynomial of K is a non-trivial product of cyclotomic polynomials if \Sigma_n(K) is an L-space for some n = 2, 3, 4, 5. Our results allow us to calculate the smooth and topological 4-ball genera of, for instance, quasi-alternating oriented quasipositive links. They also allow us to classify strongly quasipositive 3-strand pretzel knots. 

 

INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Michel Boileau (Université de Provence Aix Marseille 1)
3-manifold groups epimorphisms and rigidity

Co-Author Stefan Friedl (Universität Regensburg)

Given a degree-one map  between two aspherical compact orientable 3-manifolds it is a natural question to find out extra conditions to ensure that it is in fact homotopic to a homeomorphism. We will give two criteria in term of virtual ranks and of virtual heegaard genera. We will also discuss the knot spaces case and some open questions.

INI 1
12:30 to 13:30 Lunch @ Wolfson Court
14:00 to 15:00 Duncan McCoy (University of Texas at Austin)
Characterizing slopes for torus knots
We say that   is a characterizing slope for a knot   in the 3-sphere if the oriented homeomorphism type of  -surgery is sufficient to determine the knot   uniquely. I will discuss the problem of determining which slopes are characterizing for torus knots, paying particular attention to non-integer slopes. This problem is related to determining which knots in     have Seifert fibered surgeries.
INI 1
15:00 to 15:30 Afternoon Tea
15:30 to 16:30 Brendan Owens (University of Glasgow)
A Gordon-Litherland form for ribbon surfaces
Co-authors: Josh Greene and Saso Strle

I will describe work in progress aimed at generalising the Gordon-Litherland form to the case of properly embedded surfaces in the four-ball and the four-sphere.  I will also discuss a motivating non-application involving concordance invariants of links which lie on definite surfaces in S^4.


INI 1
19:30 to 22:00 Formal Dinner at Emmanuel College
Friday 3rd February 2017
09:00 to 10:00 Ciprian Manolescu (University of California, Los Angeles)
Floer homology and covering spaces
Co-author: Tye Lidman (North Carolina State)
 
I will discuss a Smith-type inequality for regular covering spaces in monopole Floer homology. Using the monopole Floer / Heegaard Floer correspondence, it follows that if a 3-manifold Y admits a p^n-sheeted regular cover that is a Z/p-L-space (for p prime), then Y is a Z/p-L-space. Further, we obtain constraints on surgeries on a knot being regular covers over other surgeries on the same knot, and over surgeries on other knots. This is joint work with Tye Lidman.
INI 1
10:00 to 11:00 Matthew Stoffregen (University of California, Los Angeles)
Pin(2)-equivariant Floer homology and homology cobordism
We review Manolescu's construction of the  -equivariant Seiberg-Witten Floer stable homotopy type, and apply it to the study of the 3-dimensional homology cobordism group. We introduce the `local equivalence' group, and construct a homomorphism from the homology cobordism group to the local equivalence group. We then apply Manolescu's Floer homotopy type to obstruct cobordisms between Seifert spaces. In particular, we show the existence of integral homology spheres not homology cobordant to any Seifert space. We also introduce connected Floer homology, an invariant of homology cobordism taking values in isomorphism classes of modules.
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Jen Hom (Georgia Institute of Technology)
Knot concordance in homology spheres
The knot concordance group C consists of knots in S^3 modulo knots that bound smooth disks in B^4. We consider C_Z, the group of knots in homology spheres that bound homology balls modulo knots that bound smooth disks in a homology ball. Matsumoto asked if the natural map from C to C_Z is an isomorphism. Adam Levine answered this question in the negative by showing the map is not surjective. We show that the image of C in C_Z is of infinite index. This is joint work with Adam Levine and Tye Lidman.
INI 1
12:30 to 13:30 Lunch @ Wolfson Court
14:00 to 15:00 John Baldwin (Boston College); (Boston College)
Stein fillings and SU(2) representations
Co-author: Steven Sivek (Imperial College)

I'll describe recent work with Sivek in which we prove that if a 3-manifold Y admits a Stein filling which is not a homology ball then its fundamental group admits a nontrivial SU(2) representation. Beyond establishing a new connection between contact geometry and the fundamental group, this result allows us to deduce the existence of nontrivial representations where previously existing methods do not appear to suffice. Our proof makes use of a fairly new invariant of contact 3-manifolds which Sivek and I defined in the context of instanton Floer homology.
INI 1
15:00 to 15:30 Afternoon Tea
15:30 to 16:30 Gordana Matic (University of Georgia)
Filtering the Heegaard Floer contact invariant
Co-authors: Cagatay Kutluhan ( University at Buffalo), Jeremy Van Horn-Morris (University of Arkansas), Andy Wand (University of Glasgow)

In this joint work with Kutluhan, Van Horn-Morris and Wand, we define the {\it spectral order} invariant of contact structures in dimension three by refining the contact invariant from Heegaard Floer homology. This invariant takes values in the set \mathbb{Z}_{\geq0}\cup\{\infty\}. It is zero for overtwisted contact structures, \infty for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable from any supporting open book decomposition. It gives a criterion for tightness of a contact structure stronger than that given by the Heegaard Floer contact invariant, and an obstruction to existence of Stein cobordisms between contact 3-manifolds. We show this by exhibiting an infinite family of examples with vanishing Heegaard Floer contact invariant on which our invariant assumes an unbounded sequence of finite and non-zero values.
INI 1
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons