3manifold workshop
Monday 30th January 2017 to Friday 3rd February 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) Polytope invariants of groups and manifolds
Coauthors: Wolfgang L\"uck (University of Bonn), Kevin Schreve (University of Michigan), Stephan Tillmann (University of Sydney) We will associate to L^2acyclic 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 2generator 1relator 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 UrbanaChampaign) Floer homology, group orders, and taut foliations of hyperbolic 3manifolds
A bold conjecture of BoyerGordenWatson and others posit
that for any irreducible rational homology 3sphere M the following three
conditions are equivalent: (1) the fundamental group of M is leftorderable,
(2) M has nonminimal Heegaard Floer homology, and (3) M admits a coorientable
taut foliation. Very recently, this conjecture was established for all graph
manifolds by the combined work of BoyerClay and
HanselmanRasmussenRasmussenWatson. I will discuss a computational survey of
these properties involving half a million hyperbolic 3manifolds, 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) Lspace surgeries on iterated satellites by torus links
Satellites by torus links provide multicomponent versions of
cables. I will describe the region of Lspace surgeries on any toruslink satellite of any Lspace knot, with a result that precisely extends Hedden’s and Hom’s analogous result for cables. More generally, I will characterize the region of Lspace surgeries for iterated toruslink 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 
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 etainvariant and can be expressed as a
quotient of Conway polynomials. It is also related to the correction term in
Wall’s nonadditivity formula for the signatures of 4manifolds, 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) Diagram Uniqueness for Highly Twisted Plats
Coauthor: 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 2mplat 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 biorderability of 3manifold groups
Coauthor: Kimihiko Motegi
(Nihon University)
It is known that a biorderable 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 3manifolds, and verify the conjecture for nonhyperbolic, geometric 3manifolds. We also confirm the conjecture for some infinite families of closed hyperbolic 3manifolds. 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 nonvanishing 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 TemperleyLieb category in operator algebras and in link homology
The TemperleyLieb 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 TemperleyLieb category for subfactors led to the creation of
the Jones polynomial, and thence, after a long gestation, Khovanov homology.

INI 1 
09:00 to 10:00 
Yi Ni (CALTECH (California Institute of Technology)) Null surgery on knots in Lspaces
Coauthor: 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 Lspace, 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 Lspace knots
Until July 2014, all known Lspaces admitted an involution. Then, through a clever search of the SnapPy census, DunfieldHoffmanLicata found examples of asymmetric onecusped hyperbolic manifolds with two lens space fillings and consequently many asymmetric Lspace fillings. Yet since none of these lens space fillings were $S^3$, so still stood the conjecture that Lspace knots in $S^3$ are strongly invertible. In this talk we present (1) a `natural' realization and vast generalization of the DunfieldHoffmanLicata examples (joint work with Hoffman and Licata) and (2) the first construction of asymmetric Lspace knots in $S^3$ (joint work with Luecke). Both of these constructions produce asymmetric onecusped 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) Boundaryreducing 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 
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 Lspaces
Coauthors: Michel Boileau
(Université AixMarseille), Cameron McA. Gordon
(University of Texas at Austin)
We show that if L is an oriented nontrivial strongly quasipositive link or an oriented quasipositive link which does not bound a smooth planar surface in the 4ball, then the Alexander polynomial and signature function of L determine an integer n(L) such that \Sigma_n(L), the nfold cyclic cover of S^3 branched over L, is not an Lspace for n > n(L). If K is a strongly quasipositive knot with monic Alexander polynomial such as an Lspace knot, we show that \Sigma_n(K) is not an Lspace for n \geq 6 and that the Alexander polynomial of K is a nontrivial product of cyclotomic polynomials if \Sigma_n(K) is an Lspace for some n = 2, 3, 4, 5. Our results allow us to calculate the smooth and topological 4ball genera of, for instance, quasialternating oriented quasipositive links. They also allow us to classify strongly quasipositive 3strand pretzel knots. 
INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Michel Boileau (Aix Marseille Université) 3manifold groups epimorphisms and rigidity
CoAuthor Stefan Friedl (Universität Regensburg) Given a degreeone map between two aspherical compact orientable 3manifolds 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 3sphere 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
noninteger 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 GordonLitherland form for ribbon surfaces
Coauthors: Josh Greene and Saso Strle I will describe work in progress aimed at generalising the GordonLitherland form to the case of properly embedded surfaces in the fourball and the foursphere. I will also discuss a motivating nonapplication 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 
09:00 to 10:00 
Ciprian Manolescu (University of California, Los Angeles) Floer homology and covering spaces
Coauthor: Tye Lidman
(North Carolina State)
I will discuss a Smithtype inequality for regular covering spaces in monopole Floer homology. Using the monopole Floer / Heegaard Floer correspondence, it follows that if a 3manifold Y admits a p^nsheeted regular cover that is a Z/pLspace (for p prime), then Y is a Z/pLspace. 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
SeibergWitten Floer stable homotopy type, and apply it to the study of
the 3dimensional 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) Stein fillings and SU(2) representations
Coauthor: Steven Sivek
(Imperial College)
I'll describe recent work with Sivek in which we prove that if a 3manifold 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 3manifolds 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
Coauthors: Cagatay Kutluhan ( University at Buffalo), Jeremy Van HornMorris (University of Arkansas), Andy Wand (University of Glasgow) In this joint work with Kutluhan, Van HornMorris 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, nondecreasing 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 3manifolds. 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 nonzero values. 
INI 1 