# PROGRAM

Time | Monday | Tuesday | Wednesday | Thursday | Friday |
---|---|---|---|---|---|

8:30 - 9:00 | Welcome |
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9:00 - 10:00 | Abigail Thompson | Fico González |
Free day |
Maggy Tomova | Michel Boileau |

10:15 - 11:15 | Masakazu Teragaito | Ken Baker | Scott Taylor | Rachel Roberts | |

11:15 - 11:45 | Coffee break |
Coffee break |
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11:45 - 12:45 | Román Aranda | Angeles Guevara | Jennifer Schultens | Gabriela Hinojosa | |

13:00 - 15:00 | Lunch |
Lunch |
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15:00 - 16:00 | Kazuhiro Ichihara | Luis Chan | Mariel Vazquez | Jesús Rodriguez | |

16:00 - 16:30 | Coffee break |
Coffee break |
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16:30 - 17:30 | Luis Valdez | John Luecke | Kimihiko Motegi | Cameron Gordon | |

19:00 | Birthday Dinner |

# Abstracts

### Abigail Thompson (UC Davis)

**Title:** Generalized\(^2\) Property R and surgery on links

**Abstract:**
I’lll discuss questions about surgery on links which arise naturally from the trisection decomposition of 4-manifolds developed by Gay and Kirby. These links lie in \(\#^j S^1 \times S^2\) and have surgeries yielding \(\#^k S^1 \times S^2\). We can describe families of links which have such surgeries. One can ask whether all links with such surgeries lie in these families. I’ll describe what’s known for 2-component links in the 3-sphere, both in favor and opposed to the likelihood of a positive answer.

### Masakazu Teragaito (Hiroshima University)

**Title:** Generalized torsion elements and their orders

**Abstract:**
In a group, a nontrivial element \(g\) is called a generalized torsion element if some nonempty product of conjugates of \(g\) yields the identity. Although knot or link groups are torsion-free, they can contain generalized torsion elements. The existence of such an element is an obstruction for bi-orderability. Also, we can define the order of a generalized torsion element as the minimum number of its conjugates yielding the identity.
In this talk, I review known results on this topic, in particular, focusing on the order. In particular, I give the classification of knots, more generally, \(3\)-manifolds, whose fundamental groups contain a generalized torsion element of order two, the minimum possible. The talk consists of joint works with K. Himeno, T. Ito and K. Motegi.

### Román Aránda (Binghamton University)

**Title:** Notions of Hempel distance in dimension four

**Abstract:** Hempel introduced a complexity measure of a Heegaard splitting known today as the Hempel distance. This number is computed in the curve complex of a Heegaard surface. In this talk I will survey ways of extending the definition of distance of a trisected (4-manifold, surface) pair. The main idea branches out from work of Johnson using the pants complex of a surface.

### Kazuhiro Ichihara (Nihon University)

**Title:** Cosmetic surgeries on knots in the 3-sphere

**Abstract:**
A pair of Dehn surgeries on a knot is called purely (resp. chirally) cosmetic if the obtained manifolds are orientation-preservingly (resp. -reversingly) homeomorphic. It is conjectured that if a knot in the 3-sphere admits purely (resp. chirally) cosmetic surgeries, then the knot is a trivial knot (resp. a torus knot of type (2,p) or an amphicheiral knot). In this talk, after giving a brief survey on the studies on these conjectures, I will explain some recent progresses on the conjectures.

### Luis G. Valdez Sanchez (University of Texas at El Paso)

**Title:** Incompressible planar surfaces in hyperbolic link exteriors in the 3-sphere

**Abstract:**
For n>2, examples of n-component hyperbolic links L in the 3-sphere whose exterior X contains an embedded incompressible planar surface P are known, but in all cases the slope of at least one boundary component of P is meridional or integral. In this talk we discuss the construction of links in the 3-sphere whose exteriors contain an incompressible planar surface P such that each torus boundary component of the link exterior X contains exactly one boundary component of P with non-integral and non-meridional slope. These examples contradict a recent conjecture by Mario Eudave-Munoz and Makoto Ozawa.

### Francisco J. González Acuña (IMATE-Cuernavaca)

**Title:** The number of lens spaces with group \(\mathbb{Z}_n\) and associated graphs.

**Abstract:**
We give a closed formula for the number of homeomorphism types of closed 3-manifolds with fundamental group \(\mathbb{Z}_n\). It involves the Chebyshev bias. We also study a couple of graphs associated to each lens space.

### Kenneth L Baker (University of Miami)

**Title:** Handle Numbers of Knots, Links, and 3-Manifolds

**Abstract:**
A (circular) Heegaard splitting of a (sutured) 3-manifold requires a certain number of handles. We will discuss how this count depends on the 2nd homology class of the Heegaard surface and relates to some classical invariants like tunnel number and the Thurston norm. We will also explore some curious phenomena in the theory through analogies involving knots and links.

### María de los Angeles Guevara Hernández (UNAM)

**Title:** Determinant of alternating knots derived from basic polyhedra

**Abstract:**
We study the properties of alternating knots that do not have bigons in the complementary regions of their diagrams. All the alternating knots up to 12 crossing with diagrams without bigons have their determinant larger than the corresponding value for knots with bigons and the same crossing number. Further, it has been conjectured that the knot with the maximal determinant among the knots with the same crossing number does not have an alternating diagram with bigons. However, we give a family of knots without bigons whose determinant is smaller than the determinant of knots with bigons and the same number of crossings. This is joint work with Mario Eudave.

### Luis Celso Chan Palomo (UADY)

**Title:** On the Scharlemann-Wu 2-handle additions conjecture over simple 3-manifolds.

**Abstract:**M denote a compact, orientable, simple 3-manifold, with F a boundary component of genus at least two. Denote by α a slope, that is, an isotopy class of a simple closed curve on F and denote by M[α] the result of attaching a 2-handle to M along a regular neighborhood of a representative of α in F. Note that M[α] may not be simple, that is, M[α] may contain some essential spheres, tori, annuli, disks. When this happens, α is called a degenerating slope. For two slopes α and β, denote by ∆(α, β) the minimal geometric intersection number between the isotopy classes of α and β. In
1993, Scharlemann and Wu conjectured that ∆(α, β) ≤ 5 when α and β are
basic degenerating slopes. In this talk, we exhibit, as far the authors know,
the first example of a simple 3-manifold M with two separating slopes α, β on a boundary component of genus four such that M[α] and M[β] are toroidal and ∆(α, β) = 12. This provides a negative answer to Scharlemann-Wu conjecture.

### John Luecke (UT Austin)

**Title:** Spherical knots in Lens spaces

**Abstract:**
A knot in a liens space is said to be spherical if Dehn surgery on that knot produces \(S^2 X S^1\). We give restrictions to the homology classes of spherical knots.

### Maggy Haslett-Tomova (University of Central Florida)

**Title:** Genus 1 Bridge Numbers of Satellite Knots Part I

**Abstract:**
Schubert famously showed that bridge number for knots in the 3-sphere behaves as expected for composite knots as well as for satellite knots. We will discuss an alternative approach to proving these results that allows us to generalize them. Scott Taylor will discuss some of these generalizations in Part II of this talk.

### Scott Taylor (Colby College)

**Title:** Genus 1 Bridge Numbers of Satellite Knots Part II

**Abstract:**
A classical approach to constructing lower bounds on bridge number of satellite knots is to show that the companion torus can be put into a “nice” position with respect to a height function. In this talk, I’ll sketch another approach which requires much less of the companion torus and involves collapsing the satellite knot into a spatial graph with weights on the edges. Schubert-style theorems can then be recovered from the additivity properties of bridge number.

### Jennifer Schultens (UC Davis)

**Title:** Flipping Heegaard splittings

**Abstract:**
Heegaard splittings divide a 3-manifold into two symmetric pieces called handlebodies. An interesting question pertains to understanding when these two handlebodies can be interchanged via an isotopy. The process of interchanging handlebodies is called ``flipping". This talk will explore manifolds with Heegaard splittings that can be flipped and manifolds with Heegaard splittings that can't be flipped.

### Mariel Vázquez (UC Davis)

**Title:** Results in low-dimensional topology inspired by questions in molecular biology

**Abstract:** Band surgeries and crossing changes are common in molecular biology. Certain proteins called enzymes recognize two short segments of DNA, introduce one or two breaks, and rearrange the segments before resealing the breaks. Such local rearrangements may result in global topological changes that can affect the health of an organism. This is further complicated by the fact that nucleic acids such as DNA and RNA are tightly compacted in their natural environment. We use techniques from knot theory and low-dimensional topology, aided by discrete methods and computational tools, to learn about the topology of DNA and the mechanisms of enzymes acting on it. In this talk I will present some examples and illustrate how biological questions inspire new mathematical problems.

### Kimihiko Motegi (Nihon University)

**Title:** Dehn filling trivialization on a knot group

**Abstract:**
Let \(K\) be a non-trivial knot in \(S^3\), and \(K(r)\) the result of \(r\)--Dehn filling of the exterior \(E(K)\). Each \(r\)--Dehn filling trivializes elements in the normal closure of a peripheral element represented by a simple closed curve on \(\partial E(K)\) with slope \(r\). To each element \(g \in G(K)\) assign \(\mathcal{S}_K(g) = \{ r \in \mathbb{Q} \mid r\textrm{--Dehn filling trivializes}\ g \} \subset \mathbb{Q}\). The Property P conjecture asserts that \(\mathcal{S}_K(\mu) = \emptyset\) for the meridian \(\mu\) of \(K\). In this talk we will discuss which subset \(\mathcal{R} \subset \mathbb{Q}\) can be realized as \(\mathcal{S}_K(g) = \mathcal{R}\) for some element \(g \in G(K)\). This is joint work with Tetsuya Ito and Masakazu Teragaito.

### Michel Boileau (Aix-Marseille University)

**Title:** On Symmetric union

**Abstract:**
The symmetric union is a construction introduced by Kinoshita and Terasaka in 1957 and generalized by Lamm in 2000. This construction is a generalization of the connected sum of a knot with its mirror image which produces plenty examples of ribbon knots. It is still an open question whether every ribbon knot admits a symmetric union presentation. In these talks we will discuss this question and the relationship between the knot type of a symmetric union and of its partial knot. We will also discuss some finiteness result wih respect to the partial knots. This is a joint work with Teruaki Kitano and Yuta Nozaki.

### Rachel Roberts (Washington University in St Louis)

**Title:** Taut foliations in alternating link complements

**Abstract:**
A knot is called persistently foliar if every boundary slope except one is strongly realized by a taut foliation. We discuss a construction of taut foliations in alternating link complements and prove that every nontorus alternating knot is persistently foliar.

### Gabriela Hinojosa (Universidad Autónoma del Estado de Morelos)

**Title:** Two non-equivalent families of wild knots obtained via the action of Kleinian groups

**Abstract:**
We construct two different families of wild knots. The first one is obtained as the limit set of Kleinian groups; more precisely, let \(T\) be * a pearl necklace* consists of the union of \(n\) consecutive tangent closed 3-balls (pearls) \(B_i\) (\(i=1,2,\ldots, n\)) and consider the Kleininan group \(\Gamma_T\) generated by the reflections on the boundaries \(\partial B_i\). Then the limit set \(\Lambda(\Gamma_T)\) of \(\Gamma_T\), is a wild knot in the sense of Artin and Fox which is called * a dynamically defined wild knot*.
We modify the previous construction to get the second familiy. Let \(K\) be a tame knot and consider an \(n\) dotted pearl necklace \(T^{\circ}\) which is the union of \(n\) consecutive disjoint closed round balls (pearls) \(B_j\), \(j=1,\ldots, n\). An \(n\) * pearl chain necklace* \(T\) is the union of \(T^{\circ}\) and \(K\). We will obtain, via the action of a Kleinian group, a sequence of nested pearl chain necklaces \(T_k\) whose inverse limit is * a wild knot of dynamically defined type* \(\Lambda (\Gamma_T)\).
In this talk, we will describe some topological properties of these two families of wild knots; in particular, for the first family we will study their Hausdorff dimensions. For the second family, we generalize the construction of cyclic branched coverings, and we will show that there exists a wild knot of dynamically defined type such that \(\mathbb{S}^3\) is an \(n\)-fold cyclic branched covering of \(\mathbb{S}^3\) branched over it, for \(n\geq 2\).

### Jesús Rodriguez-Viorato (CONAHCyT-CIMAT)

**Title:** A bound on the number of twice-punctured tori in a knot exterior

**Abstract:**We continue a program due to Motegi regarding universal bounds for the number of non-isotopic essential $n$-punctured tori in the complement of a hyperbolic knot in $S^3$. For $n=1$, Valdez-S\'anchez showed that, at most, five non-isotopic Seifert tori exist in the exterior of a hyperbolic knot. In this talk, we address the case $n=2$.

### Cameron Gordon (University of Texas at Austin)

**Title:** Cyclic branched covers of links and the L-space Conjecture

**Abstract:**
The L-space Conjecture asserts that for a prime 3-manifold \(M\) the following are equivalent: \(M\) is not a Heegaard Floer L-space, \(\pi_1(M)\) is left-orderable, and \(M\) admits a co-orientable taut foliation. We will discuss these three properties for cyclic branched covers of knots and links.
This is joint work with Steve Boyer and Ying Hu.