Lowdimensional topology on Skye
The theme of this workshop will be the topology of smooth lowdimensional manifolds. A central topic will be the conjectural relationship
between Heegaard Floer Lspaces, leftorderability, and taut foliations. A range of other topics will also be explored.
Format and other details

This will be a Banffstyle workshop. Accommodation and meals onsite will be provided. We will also provide a
coach from Glasgow to Skye and back
(details below).
Some funds for travel to Glasgow may be available for those who require them.
A schedule of talks will be posted shortly before the beginning of the workshop.
Travel to Skye

We have booked a coach which will take participants from Glasgow to Skye on Sunday 11th June.
It will depart from Wolfson Medical Building, University Avenue
at 3:30pm and will then make a stop at Glasgow Airport (turn right out of terminal building for coach park) at 4pm.
There will be a refreshment stop for about 45 minutes at Fort William at around 6:30pm, which will give a chance to get some food.
We expect to arrive at SMO around 9:30pm.
The return coach will leave Skye for Glasgow after breakfast on Saturday 17th June. We expect to arrive at
Glasgow Airport between 3pm and 4pm.
Alternatives: if the coach does not suit you, you may consider renting a car and driving;
or taking the train from Glasgow Queen St, to connect with the MallaigArmadale ferry. The train
and ferry timetable is here.
From certain locations it may be possible to take a train to Kyle of Lochalsh, which is within taxi distance from the conference.
For train times in the UK see http://www.nationalrail.co.uk.
For more useful information on travel to Skye, such as where to land your helicopter, try this link,
which is from the web page of a different conference.
Schedule
(Titles and abstracts below)
Participants
 Paolo Aceto, Renyi Institute
 Ken Baker, Miami
 John Baldwin, Boston College
 Hans Boden, McMaster
 Steve Boyer, UQàM
 Danny Calegari, Chicago
 Vincent Colin, Nantes
 Vaibhav Gadre, Glasgow
 Cameron Gordon, Texas
 Josh Greene, Boston College
 Chris Herald, UNR
 Kyle Larson, MSU
 Yankı Lekili, KCL
 Adam Levine, Princeton
 Joan Licata, ANU
 Tony Licata, ANU
 Francesco Lin, Princeton
 Robert Lipshitz, Oregon
 Andrew Lobb, Durham
 Joseph MacColl, UCL
 Gordana Matić, Georgia
 Irena Matkovič, CEU
 Duncan McCoy, Texas
 Allison Miller, Texas
 Matthias Nagel, McGill
 Brendan Owens, Glasgow
 Luisa Paoluzzi, Marseilles
 Lisa Piccirillo, Texas
 Sarah Rasmussen, Cambridge
 Rachel Roberts, WSUL
 John Shareshian, WSUL
 Steven Sivek, MPIM
 Mike Snape, Glasgow
 András Stipsicz, Renyi Institute
 Matthew Stoffregen, UCLA
 Sašo Strle, Ljubljana
 Daniel Waite, Glasgow
 Andy Wand, Glasgow
 Liam Watson, Glasgow/Sherbrooke
 Claudius Zibrowius, Cambridge
Talks

Monday June 12
Leftorderability, foliations, Lspaces and cyclic branched covers, Cameron Gordon
It has been conjectured that for a prime 3manifold M the following are equivalent: (1) π_{1}(M) is leftorderable, (2) M admits a coorientable taut foliation, and (3) M is not a Heegaard Floer Lspace. We will discuss these properties in the special case where M is the nfold cyclic branched cover of a knot.
Morse Structures on Open Books, Joan Licata
Every contact 3manifold is locally contactomorphic to the standard contact R^{3}, but this fact does not necessarily produce large charts that cover the manifold efficiently. I'll describe joint work with Dave Gay and more recently, Dan Mathews, 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 3manifold from these charts. We use this construction to define front projections for Legendrian knots and links in arbitrary contact 3manifolds, generalising existing constructions of front projections for Legendrian knots in S^{3} and universally tight lens spaces.
Mirror symmetry for punctured surfaces and Auslander orders, Yankı Lekili
We consider partially wrapped Fukaya categories of punctured surfaces with stops at their boundary. We prove equivalences between such categories and derived categories of modules over the Auslander order on certain nodal stacky curves. As an application, we deduce equivalences between derived categories of coherent sheaves (resp. perfect complexes) on such nodal stacky curves and the wrapped (resp. compact) Fukaya categories of punctured surfaces of arbitrary genus. This is joint work with Polishchuk.
Features of knots that are sometimes related to the presence of symmetries, Luisa Paoluzzi
Symmetric knots are obviously special. After recalling what symmetries are and presenting some basic properties, I will survey three types of situations in which the presence of symmetries may be related to other characteristic of the knot, namely its behaviour with respect to orientations, to cyclic branched cover, and to the fact of being trivial.
(1,1) Lspace knots, Josh Greene
I will prove and discuss a characterization of (1,1) Lspace knots in joint work with Sam Lewallen and Faramarz Vafaee.

Tuesday June 13
Constructing CTFs, Rachel Roberts
I will describe a construction of (codimension one) cooriented taut foliations (CTFs) of 3manifolds. It follows from this construction that if K is a composite, alternating,
or Montesinos knot, then the Lspace conjecture of Ozsváth and Szabó holds for any 3manifold obtained by Dehn surgery along K. This work is joint with Charles Delman.
Bordered involutive Floer homology, Robert Lipshitz
HendricksManolescu introduced involutive Heegaard Floer homology as a partial analogue of pinequivariant monopole Floer homology. We will discuss an algorithm for computing involutive HFhat using bordered Floer homology and a proposed definition of bordered involutive Floer homology. This is joint work with Kristen Hendricks.
On peculiar modules for 4ended tangles, Claudius Zibrowius
A peculiar module is a certain invariant of 4ended tangles that I developed in my PhD thesis as a tool for studying the local behaviour of Heegaard Floer homology for knots and links. I will briefly explain its construction and then discuss two or three features of interest, such as mutation symmetries, skein relations and the role of the Fukaya category of the 4punctured sphere.
 Wednesday June 14
Universal Circles, Danny Calegari
Braid groups and categorical actions, Tony Licata
Braid groups (and, more generally, Artin groups of Coxeter groups) arise naturally in modern representation theory as
"categorical reflection groups" acting on triangulated categories by "categorical reflections." The goal of this talk will be to explain how to see various structures of interest in the study of braid groups using their appearance as categorical reflection groups.
Knot traces and concordance, Allison Miller
A conjecture of Akbulut and Kirby from 1978 states that the concordance
class of a knot is determined by its 0surgery. In 2015, Yasui disproved
this conjecture by providing pairs of knots which have the same
0surgeries yet which can be distinguished in (smooth) concordance by an
invariant associated to the fourdimensional traces of such a surgery.
In this talk, I will discuss joint work with Lisa Piccirillo in which we
construct many pairs of knots which have diffeomorphic 0surgery traces
yet some of which can be distinguished in smooth concordance by the
Heegaard Floer dinvariants of their double branched covers. If time
permits, I will also discuss the applicability of this work to the
existence of interesting invertible satellite maps on the smooth
concordance group.
PseudoAnosov maps with small entropy and the curve complex, Vaibhav Gadre
The talk will survey the theory of pseudoAnosov maps with small entropy subsequently focussing on deriving bounds in terms of genus for small translation distances in the curve complex. The main result is joint work with Chiayen Tsai
Orderability questions in contact geometry, Vincent Colin
I will review various results regarding orderability of the group of contactomorphisms of a contact manifolds and of the space of Legendrian submanifolds. This includes a work of Liu as well as joint work with FerrandPushkar and ChantraineDimitroglou Rizell.
 Thursday June 15
A topological invariant for left orders, Sarah Rasmussen
I will discuss work in progress involving a
topological invariant for left invariant orders
on the fundamental group of a 3manifold. This
invariant measures an obstruction for
producing a taut foliation from a left order.
An Odd Khovanov Homotopy Type, Matt Stoffregen
We define an odd Khovanov homotopy type, in analogy with the version of the (even) Khovanov homotopy type constructed by LawsonLipshitzSarkar, and list some of its basic properties, as well as conjectural relationships with other invariants. This is work in progress with Sucharit Sarkar and Chris Scaduto.
Non Lspaces and spectral geometry, Francesco Lin
We discuss an application of monopole Floer homology to the spectral geometry of threemanifolds: on a rational homology sphere which is not an Lspace, for every Riemannian metric the first eigenvalue of the Laplacian on coexact one forms is bounded above very explicitly in terms of the Ricci curvature.

Friday June 16
Satellite Lspace knots are braided satellites, Ken Baker
Let {K_{n}} be the family of knots obtained by twisting a knot K along an unknot c.
When the winding number of K about c is nonzero, we show the limit of g(K_{n})/g_{4}(K_{n}) is 1 if and only if the winding and wrapping numbers of K about c are equal. When equal, this leads to a description of minimal genus Seifert surfaces of K_{n} for n>>0 and eventually to a characterization of when c is a braid axis for K. We then use this characterization to show that satellite Lspace knots are braided satellites. This is joint work with Kimihiko Motegi that builds upon joint work with Scott Taylor.
Knots with infinitely many SU(2)cyclic surgeries, Steven Sivek
The cyclic surgery theorem of Culler, Gordon, Luecke, and Shalen implies that any knot in S^{3} other than a torus knot has at most two nontrivial cyclic surgeries. In this talk, we investigate the weaker notion of SU(2)cyclic surgeries on a knot, meaning surgeries whose fundamental groups only admit SU(2) representations with cyclic image. We will show that if a nontrivial knot in S^{3} has infinitely many SU(2)cyclic surgeries, then the corresponding slopes (viewed as a subset of RP^{1}) have a unique limit point, which is a finite, rational number, and that this limit is a boundary slope for the knot. As a corollary, it follows that for any nontrivial knot, the set of SU(2)cyclic surgery slopes is bounded. This is joint work with Raphael Zentner.
Khovanov homology detects the trefoil, John Baldwin
In 2010, Kronheimer and Mrowka proved that Khovanov homology detects the unknot, answering a "categorified" version of the famous open question: Does the Jones polynomial detect the unknot? An even more difficult question is: Does the Jones polynomial detects the trefoils? The goal of this talk is to outline our proof that Khovanov homology detects the trefoils, answering a "categorified" version of this second question. Our proof, like Kronheimer and Mrowka's, relies on a relationship between Khovanov homology and instanton Floer homology. More surprising, however, is that it also hinges fundamentally on several ideas from contact and symplectic geometry. This is joint work with Steven Sivek.
Filtering the Heegaard Floer contact invariant, Gordana Matić
We will explore a filtration on a stabilized Heegaard Floer complex associated with an open book decomposition compatible with a contact structure to define a new contact invariant "spectral order" which refines the OzsvathSzabo contact class. The definition is partially motivated by Hutcthings' interpretation of Algebraic Torsion of Latchev and Wendl. The spectral order is zero when the structure is over twisted, infinite when it is Stein fillable, and is often nonzero and gives us more information when the OzsvathSzabo contact invariant is zero but the contact structure is tight.
This is joint work with Cagatay Kutluhan, Jeremy Van HornMorris
and Andy Wand.
Concordance of knots in homology spheres, Adam Levine
Every knot in the 3sphere bounds a nonlocally flat piecewiselinear (PL) disk in the 4ball, but Akbulut showed in 1990 that the same is not true for knots in the boundary of an arbitrary contractible 4manifold. We strengthen this result by showing that there exists a knot K in a homology sphere Y (which is the boundary of a contractible 4manifold) such that K does not bound a PL disk in any homology 4ball bounded by Y. In more recent work (joint with Jen Hom and Tye Lidman), we show that the group of knots in homology spheres modulo nonlocallyflat PL concordance contains an infinite cyclic subgroup.
Funding

We gratefully acknowledge support from the European Commission (Marie Curie career integration grant HFFUNDGRP) and the School of Mathematics and Statistics at the University of Glasgow.