# The 5th East Asian School of Knots and Related Topics

## Abstracts

 Yinghua Ai (Peking University) Applications of twisted Floer homology (joint work with Yi Ni and with Thomas Peters) Using Heegaard Floer homology with twisted coefficients in the universal Novikov ring, we gave a necessary and sufficient condition for an irriducible closed 3-manifold to be a torus bundle over the circle. Slides: [YinghuaAi.pdf]

 Byung Hee An (KAIST) A family of permutation braids whose reduced super summit set grows exponentially The best known algorithm for conjugacy problem is to generate the reduced super summit set after taking a suitable power of given braid to make it rigid. We first present an inductive algorithm that generates rigid permutation braids with a large reduced super summit set whose growth can be verified to be exponential by experiment. Moreover we give another family of rigid permutation braids whose reduced super summit set can be shown to grows exponentially, and conclude that the conjugacy problem remains exponential in braid index under the current knowledge. Slides: [ByungHeeAn.pdf]

 Yongju Bae (Kyungpook National University) On the Bollobas-Riordan polynomial of certain ribbon graphs Bollobas-Riordan polynomials of ribbon graphs are closely related with link invariants(e.g. Homply polynomial, Kauffman Bracket, etc). In this talk, we will construct a composition of two ribbon graphs, which corresponds to the sum of 4-tangles in knot theory, and give a formula to calculate its Bollobas-Riordan polynomial in virtue of Bollobas-Riordan polynomials of its factors. Slides: [YongjuBae.pdf]

 Zhiqiang Bao (Peking University) Manifolds associated with $(Z_2)^n$-colored regular graphs We will describe a way to expand $(Z_2)^n$-colored regular graphs into closed manifolds by adding cells determined by the edge-coloring inductively. Every closed combinatorial manifold can be obtained in this way. In addition, if a $(Z_2)^n$-colored regular graph admits this construction, then it is realizable as the moment graph of a closed $n$-manifold with $(Z_2)^n$-action. Slides: [ZhiqiangBao.pdf]

 Jason Behrstock (Lehman College, City University of New York) Geometry and rigidity of mapping class groups We will discuss the geometry of the mapping class group and applications to quasi-isometric rigidity. This will include the recent result that any quasi-isometry of the mapping class group is a bounded distance away from an isometry induced by left multiplication. This has the consequence that any finitely generated group which is quasi-isometric to the mapping class groups is, up to finite groups, isomorphic to it. This talk will be accessible to a broad audience of geometers and topologist and will include a survey of the background and history of these results. Slides: [JasonBehrstock.pdf]

 Jae Choon Cha (POSTECH) Local derived series, L2-signatures, and applications to manifolds Aimed to geometric applications, we introduce a new commutator-type series of groups associated to homology localizations of groups and rings due to Bousfield, Vogel, and Cohn. The series, called the local derived series, provides both functoriality and an injectivity theorem and refines Harvey's torsion-free derived series. We give some applications to distinct homology cobordism types within the same simple homotopy type in higher dimensions, and to 3-dimensional spherical space forms. As a key ingredient for the applications, we prove the homology cobordism invariance of L2-betti numbers and L2-signature defects associated some interesting infinite/finite non-torsion-free amenable groups. This is joint work with Kent Orr. Slides: [JaeChoonCha.pdf]

 Fan Ding (Peking University) Contact Dehn surgery and its applications This is a joint work with Hansjörg Geiges. Every closed, connected contact 3-manifold can be obtained from the 3-sphere with its standard contact structure by contact (±1)-surgery along a Legendrian link. In combination with Heegaard Floer theory, this surgery presentation theorem has turned out to be a powerful tool (in the hands of Lisca and Stipsicz, for instance) for the classification of contact 3-manifolds. We also described various handle moves in contact surgery diagrams, notably contact analogues of the Kirby moves. Slides: [FanDing.pdf]

 Kun Du (Dalian University of Technology) Unstabilized self-amalgamations Let M be an orientable closed 3-maninifold, F is a non-separating closed surface. Let M' be the 3-manifold obtained by cutting M along F. In this paper, we prove that if M' has a high distance Heegaard splitting, then M has a unique minimal Heegaard splitting up to isotopy. Slides: [KunDu.pdf]

 Stefan Friedl (University of Warwick) Twisted Alexander polynomials and fibered 3-manifolds It is a classical result that for a fibered knot the Alexander polynomial is monic and that the degree of the Alexander polynomial is determined by the genus. These fibering obstructions have been generalized to twisted Alexander polynomials by Cha, Goda-Kitano-Morifuji, Friedl-Kim and Kitayama. In this talk I will show that these are in fact complete fibering obstructions, i.e. twisted Alexander polynomials detect fibered 3-manifolds. This talk is based on joint work with Stefano Vidussi. Slides: [StefanFriedl.pdf]

 Matt Hedden (MIT/Indiana U./Michigan State U.) Applications of numerical invariants from the knot Floer homology filtration The knot Floer homology invariants of Ozsvath and Szabo, and Rasmussen, take the form of a filtered chain complex associated to a knot in a three-manifold. This allows one to define numerical invariants of the knot. I will discuss applications of these invariants to the study of classical knot and link concordance, the homology concordance group of knots in homology spheres, the theory of Legendrian and transverse knots in contact manifolds, and the interaction between knot theory and complex curves (resp. symplectic surfaces) in Stein domains (resp. symplectic 4-mflds) with boundary. Slides: [MattHedden.pdf]

 Ryuji Higa (Kobe University) Tangle sum of alternating tangles yielding a trivial knot We consider the problem to decide whether a given diagram represents a trivial knot or not, by using Menasco's crossing ball argument. This problem is solved in the cases of alternating diagrams and of almost alternating ones by Menasco-Thistlethwaite and by Tsukamoto. In this talk, we study what kind of tangle sum of alternating tangles yielding a trivial knot. Slides: [RyujiHiga.pdf]

 Youngsik Huh (Hanyang University) An upper bound on stick numbers of knots In this talk we give an upper bound on stick number of knots in terms of arc index, by which the Negami's upper bound on stick number is slightly improved. This is cowork with Seungsang Oh. Slides: [YoungsikHuh.pdf] [YoungsikHuh_Knot Table Search Results.mht]

 Ayumu Inoue (Tokyo Institute of Technology) A natural ch-diagram of the n-twist-spun trefoil knot I have gotten a ch-diagram of the $n$-twist spun trefoil knot by constructing a Computer Graphics of the twist spun trefoil knot and slicing it. I would like to introduce this ch-diagram and potentials of Computer Graphics to visualize or analyze surface knots. Slides: [AyumuInoue.pdf] [AyumuInoue_00_red_blue_glasses_en.png] [AyumuInoue_motion_picture_1twist.avi] [AyumuInoue_motion_picture_spun_grow.avi] [AyumuInoue_sheet_st.avi]

 Atsushi Ishii (University of Tsukuba) Flows for spatial graphs and handlebody-knots A spatial graph is a graph embedded in the 3-sphere. A handlebody-knot is a handlebody embedded in the 3-sphere. We introduce a flow of a spatial graph. We see how we obtain invariants for spatial graphs and handlebody-knots by using flows. This is a joint work with Masahide Iwakiri. Slides: [AtsushiIshii.pdf]

 Tetsuya Ito (The University of Tokyo) An application of braid ordering to knot thoery Braid group has natural left-invariant total ordering, called the Dehornoy ordering. We study an application of braid ordering to knot theory, by using the Dehornoy floor, which is non-negative integer determined by the Dehornoy ordering. We show new lower bound of knot genus via Dehornoy floor, and relationships between an essential surface in closed braid complement and Dehornoy floor. Slides: [TetsuyaIto.pdf]

 Masahide Iwakiri (Osaka City University Advanced Mathematical Institute(OCAMI)) Singular surface-links with braid index 3 In this talk, we show that the $w$-index of a singular surface-link with braid index $3$ is $0$, which was proved for embbeded one by S. Kamada. As consequences, we also prove that both of unknotting numbers of a singular surface-link with braid index $3$ associated with $1$-handle surgeries and crossing changes are at most $1$. Slides: [MasahideIwakiri.pdf]

 Yeonhee Jang (Hiroshima University) A classification of 3-bridge algebraic links An algebraic link is a link obtained by summing rational tangles together. We give a classification of 3-bridge algebraic links up to isotopy. Moreover, for non-Montesinos ones, we also give a classification of 3-bridge spheres of each link up to isotopy. Slides: [YeonheeJang.pdf]

 Boju Jiang (Peking University) A remark on fixed points of graph maps Consider self-maps of a graph (= finite 1-dimensional cell complex). Bestvina and Handel in their famous proof of the "Scott Conjecture" established some inequality relating the "rank" of fixed point classes. Later, an inequality relating the index of fixed point classes has been proved. We now show a new inequality which unifies and sharpens the previous two. This is joint work with Shida Wang and Qiang Zhang. Slides: [BojuJiang.pdf]

 In Dae Jong (Osaka City University) Seifert fibered surgeries on Montesinos knots (joint work with Kazuhiro Ichihara and Shigeru Mizushima) In this talk, Dehn surgeries on hyperbolic Montesinos knots yielding Seifert fibered spaces will be discussed. Actually we will present a complete classification of the Dehn surgeries on hyperbolic Montesinos knots which yield manifolds with cyclic or finite fundamental groups. We will also give certain restrictions on the alternating hyperbolic Montesinos knots which admit surgeries yielding Seifert fibered spaces with infinite fundamental groups. Slides: [InDaeJong.pdf]

 Teruhisa Kadokami (Dalian University of Technology) An integral invariant from the knot group (joint work with Z. Yang) For a knot $K$ in $S^3$, J. Ma and R. Qiu defined an integral invariant $a(K)$ which is the minimal number of elements that generate normally the commutator subgroup of the knot group, and showed that it is a lower bound of the unknotting number. We prove that it is also a lower bound of the tunnel number. If the invariant were addivive under connected sum, then we could deduce something about additivity of both the unknotting numbers and the tunnel numbers. However we found a sequence that the invariant is not additive under connected sum as follows : Let $T(2, p)$ be a torus knot of type $(2, p)$, and $K(p, q)=T(2, p)¥sharp T(2, q)$. Then we find sequences of $¥{K(p, q)¥}$ that $a(K(p, q))=1$ holds. Slides: [TeruhisaKadokami.pdf]

 Seiichi Kamada (Hiroshima University) Linear biquandles and long virtual knots For a ring $R$ and a (2,2)-matrix $A$ over $R$, we define four $R$-modules associated with a long virtual knot diagram. If the ring and the matrix satisfy a certain condition then the modules are invariants of long virtual knots. We also consider how to get polynomial invariants from the modules. This is an introduction of the paper New invariants of long virtual knots' by A. Bartholomew, R. Fenn, N. Kamada and S. Kamada, to appear in the Kobe J. Math. Slides: [SeiichiKamada.pdf]

 Taizo Kanenobu (Osaka City University) On H(2)-unknotting number of a knot This is a joint work with Yasuyuki Miyazawa (Yamaguchi University). An $H(2)$-move is a local move of a knot which is performed by adding a half-twisted band. It is known an $H(2)$-move is an unknotting operation. We define the $H(2)$-{\it unknotting number} of a knot $K$ to be the minimum number of $H(2)$-moves needed to transform $K$ into a trivial knot. We give several methods to estimate the $H(2)$-unknotting number of a knot. Slides: [TaizoKanenobu.pdf]

 Akio Kawauchi (Osaka City University) The unknotting number and the warping degree of a spatial graph We introduce a topological invariant of every connected spatial graph without degree one vertex which we call the warping degree. We also generalize the unknotting number of a knot to a topological invariant of every connected spatial graph without degree one vertex, which is a relaxed invariant of the warping degree. The warping degree and the unknotting number are further used to define some analogous (but geometric) invariants of every connected spatial graph with vertices of degree one such as a knotted arc. The warping degree is also used to introduce a complexity of a (possibly disconnected) spatial graph. Use: PowerPoint for my own laptop Slides: [AkioKawauchi.ppsx]

 Jungsoo Kim (Konkuk University) Handlebody-preserving finite group actions on Haken manifolds with Heegaard genus two. Let M be a closed orientable 3-manifold of Heegaard genus two with a non-trivial JSJ-decomposition and G be a finite group of orientation-preserving smooth actions on $M$ which preserves each handlebody of Heegaard splitting and each piece of the JSJ-decomposition of M. We will call some specific conditions of the Heegaard splitting and the JSJ-tori the condition A, ... , and the condition F. In this talk, we first determine the possible isomorphism types of G when the Heegaard splitting and the JSJ-tori satisfy the condition A, the condition B, ... , or the condition F by Theorem 1.1 and Theorem 1.2. By these theorems and T. Kobayashi's results, we determine the possible isomorphism types of G when M has separating JSJ-tori or the Heegaard splitting is strongly irreducible in the Main Theorem. Slides: [JungsooKim.pdf]

 Se-Goo Kim (Kyung Hee University) Splitting property of von Neumann rho-invariants (Joint work with Taehee Kim) We give a sufficient condition for the connected sum of two given knots under which the two knots have vanishing von Neumann rho-invariants if the connected sum does. The condition is described in terms of metabolizers of Cochran-Orr-Teichner's higher-order linking forms, which includes coprimeness of Alexander polynomials. We then present an example as an application. Slides: [Se-GooKim.pdf]

 Taehee Kim (Konkuk University) The parity of degrees of higher-order Alexander polynomials for a 3-manifold For a 3-manifold there is a (never decreasing) sequence of invariants which are the degrees of the higher-order Alexander polynomials due to Cochran and Harvey. In this talk we show that any jump in the sequence is necessarily even. To show this, we use a certain Cohn localization and the duality of Reidemester torision. This is joint work with S. Friedl. Slides: [TaeheeKim.pdf]

 Yasto Kimura (The University of Tokyo) Surgeries on shadow coloured 2-diagrams Third homology classes of a quandle $$Q$$ can be represented by $$Q$$-shadow coloured 2-diagrams on some closed surfaces. In this talk, we consider some surgeries on this diagrams, and show that, if $$Q$$ is a knot quandle of a prime knot, each third homology class can be represented by a diagram on a 2-sphere, or a link diagram. Therein, we can see the importance of the "shadow fundamental classes", which are third homology classes obtained by giving shadow-colours to the regions of knot diagrams. Slides: [YastoKimura.pdf]

 Eiko Kin (Tokyo Institute of Technology) Pseudo-Anosov braids with small entropy and the magic 3-manifold We consider a hyperbolic surface bundle over the circle with the smallest known volume among hyperbolic 3-manifolds having 3 cusps, so called the magic manifold $M$. We compute the entropy function on the fiber face $\Delta \subset H_2(M, \partial M; {\Bbb R})$, determine homology classes whose representatives are genus $0$ fiber surfaces in $M$, and describe their monodromies by braids. Among homology classes whose representatives are genus $0$ fiber surfaces with $p$ punctures, we decide which one realizes the minimal entropy. It turns out that the braids which give monodromies for such homology classes have the smallest known entropy. Slides: [EikoKin.pdf]

 Kengo Kishimoto (Osaka City University) The IH-distance for spatial trivalent graphs This is a joint work with Atsushi Ishii. He showed that two spatial trivalent graphs in $S^3$ are neighborhood equivalent if and only if one can be transformed into the other by IH-moves and isotopies of $S^3$. We define the IH-distance between two spatial trivalent graphs $L$ and $L'$, denoted by $d_{IH}(L,L')$, to be the minimal number of IH-moves needed to deform $L$ into $L'$. We give a lower bound for the IH-distance by using quandle colorings for spatial graphs introduced by A.~Ishii and M.~Iwakiri. Slides: [KengoKishimoto.pdf]

 Takahiro Kitayama (The University of Tokyo) Reidemeister torsion for linear representations and Seifert surgery on knots We study an invariant of a $3$-manifold consists of Reidemeister torsion for linear representations which pass through a finite group. We show a Dehn surgery formula on these invariants and compute those of Seifert manifolds over $S^2$. As a consequence we obtain a necessary condition for a result of Dehn surgery along a knot to be Seifert fibered, which can be applied even in a case where the Alexander polynomial is trivial and abelian Reidemeister torsion gives no information. Slides: [TakahiroKitayama.pdf]

 Toshitake Kohno (The University of Tokyo) Bar complex, configuration spaces and finite type invariants for braids The purpose of this talk is to review developments in the study of configuration spaces and invariants for braids derived from iterated integrals. We show that the bar complex of the configuration space of ordered distinct points in the complex plane is acyclic. The 0-dimensional cohomology of this bar complex is identified with the space of finite type invariants for braids. We construct a universal holonomy homomorpshim from the braid group to the space of horizontal chord diagrams over the field of rational numbers, which provides finite type invariants for braids with values in rational numbers. Slides: [ToshitakeKohno.pdf]

 Eon-Kyung Lee (Sejong University) Roots in some Artin groups It is known that in the braid group on $n$ strands, the $m$-th roots of an element are unique up to conjugacy. It is also known that in the pure braid group on $n$ strands, the $m$-th roots of an element are unique. In this talk, we study what happens for partially pure braids, and then apply this result to some Artin groups. Slides: [Eon-KyungLee.pdf]

 Hwa Jeong Lee (KAIST) Arc index and nonalternating clasping Suppose that $K$ is a nonalternating knot with $n$ crossings which is obtained by clasping a reduced alternating diagram with $n-2$ crossings. We study the cases when the arc index of $K$ is equal to $n$ and when it is less than $n$. Slides: [HwaJeongLee.pdf]

 Jung Hoon Lee (KIAS) Rectangle condition for strong irreducibility of Heegaard splittings Casson and Gordon gave the rectangle condition for strong irreducibility of Heegaard splittings. We give a weak version of rectangle condition for manifolds with non-empty boundary. We give some examples including a strongly irreducible Heegaard splitting which is not a minimal genus Heegaard splitting. Slides: [JungHoonLee.pdf]

 Sang-Jin Lee (Konkuk University) Conjugacy of braids in the centralizer of a periodic braid A braid is said to be periodic if it has a central power. By Kerekjarto-Brower-Eilenberg’s Theorem, any periodic braid is represented by a rigid rotation of a punctured disk. In this talk, we show that if two n-braids in the centralizer of a periodic braid are conjugate in the n-braid group, then they are conjugate in the centralizer of the periodic braid. This implies that some monomorphisms between Artin groups induce injective functions on the set of conjugacy classes. Slides: [Sang-JinLee.pdf]

 Sangyop Lee (Chung-ang University) Boundary structure of hyperbolic 3-manifolds and exceptional Dehn fillings (joint work with Masakazu Teragaito) Let M be a hyperbolic 3-manifold with boundary a union of tori. We estimate the number of exceptional slopes on a torus boundary component of M. Slides: [SangyopLee.pdf]

 Fengling Li (Harbin Institute of technology) A sufficient condition for genera of amalgamated 3-manifolds not to go down Let Mi be a connected, compact, orientable 3-manifold, Fi a boundary component of Mi with g(Fi) greater than or equal to 2, i=1,2, and F1 is homeomorphic to F2. Let M be the surface sum of the two manifolds M1 and M2 along a homeomorphism between F1 and F2. Suppose that there is no essential bounded surface in Mi with Euler characteristic greater than 3-2g(Mi), i=1,2. Then g(M)=g(M1)+g(M2)-g(F). Slides: [FenglingLi.pdf]

 Yannan Li (Dalian Jiaotong University) 2-string free tangles and incompressible surfaces Suppose K is a connected sum of two knots, one of which admits a 2-string essential free tangle decomposition, then the exterior of K contains an incompressible surface of genus n for each positive integer n. Slides: [YannanLi.pdf]

 Ximin Liu (South China University of Technology) Group actions on 4-manifolds and Seiberg-Witten theory In this talk, some topics and results around nonsmoothable group actions on 4-manifolds are given. Especially, I will explain some recent results about nonsmoothable group actions on 4-manifolds, which are related to the Seiberg-Witten theory. Slides: [XiminLiu.pdf]

 Mineko Matsumoto (Soka University) A Partial Order on the set of Prime Knots with up to 11 Crossings Let K_1, K_2 be prime knots and G(K_1), G(K_2) their knot groups. If there exists a surjective homomorphism from G(K_1) onto G(K_2), we can define a partial order K_1 \geq K_2. In this study, we decide the partial order on the set of prime knots with up to 11 crossings. For this purpose, we need to consider 640,800 pairs of knots. Our processes are described as below: 1. We use the Alexander polynomial and the twisted Alexander polynomial to prove nonexistence of surjective homomorphism. 2. We search actually surjective homomorphism for the remaining cases. In the both processes, we use the computer to calculate the invariants and to construct surjective homomorphisms. This is a joint work with Keiichi HORIE, Teruaki KITANO and Masaaki SUZUKI. Slides: [MinekoMatsumoto.pdf]

 Yasuyuki Miyazawa (Yamaguchi University) Gordian distance and polynomial invariants We call the minimum number of crossing changes needed to transform a knot $K$ into another knot $K'$ the Gordian distance from $K$ to $K'$. In this talk, we give some criteria on the Gordian distance by using polynomial invariants. Slides: [YasuyukiMiyazawa.pdf]

 Kanji Morimoto (Konan University) Tunnel numbers of twisted torus knots with essential tori In the present talk, we will determine the tunnel numbers of some twisted torus knots with essential tori. Slides: [KanjiMorimoto.pdf]

 Hiromasa Moriuchi (Osaka City University Advanced Mathematical Institute) An enumeration of non-prime theta-curves and handcuff graphs In the 1st and the 2nd East Asian School of Knots and Related Topics, we enumerated all the prime theta-curves and handcuff graphs with up to seven crossings by using algebraic tangles and prime basic theta-polyhedra. Here, a theta-polyhedron is a connected graph embedded in a 2-sphere, whose two vertices are 3-valent, and the rest are 4-valent. We can composite many spatial graphs by using connected sum'' of them. However, for spatial graphs, `connected sum'' is not unique. Therefore we improve theta-polyhedra to enumerate non-prime theta-curves and handcuff graphs. We can obtain a theta-curve and handcuff graph diagram from a theta-polyhedron by substituting algebraic tangles for their 4-valent vertices. Slides: [HiromasaMoriuchi.pdf]

 Daniel Moskovich (RIMS, Kyoto University) An Alexander polynomial for coloured knots To obtain the Alexander polynomial of a knot K by Dehn surgery, one first obtains a surgery presentation L of K in the complement of a standard unknot U. The Alexander polynomial is equal (up to the standard indeterminancy) to the determinant of equivariant linking matrix of this surgery presentation in the infinite cyclic cover of S^3-U, which can be read "downstairs" off the winding matrix for L. I will discuss an attempt generalize this construction to obtain polynomial invariants associated to non-abelian covering spaces rather than the infinite cyclic cover. Such covering spaces arise from a transitive representation \rho from the knot group onto a non-abelian group G. When G is a dihedral group D_{2n} (joint work with A. Kricker) or the alternating group A_4, we obtain analogues of surgery presentations for the pair (K,\rho), which lift to surgery presentations of the associated covering spaces and of the covering links for K which they contain. The determinants of the associated winding matrices give polynomial invariants (up to a certain indeterminancy) for (K,\rho). Slides: [DanielMoskovich.pdf]

 Takuji Nakamura (Osaka Electro-Communication University) $C_n$-moves and periodic knots For each local move $T$, we can define the $T$-gordian distance for two knots. Our motivation is to research a relationship between the periodicity of knots and $T$-gordian distance. In this talk, we pay attention to $C_n$-move. In fact we show that for any $p$ and $n(>2)$ there exists a periodic knot of period $p$ whose $C_n$-gordian distance to the trivial knot is one. We will mention this result about other local moves. Slides: [TakujiNakamura.pdf]

 Yasutaka Nakanishi (Kobe University) On non self Delta moves (Joint work with Tetsuo Shibuya and Tatsuya Tsukamoto) We will consider a local move, which generates an equivalence relation for links satisfying the following conditions: (1) The numbers of components coincide. (2) The corresponding linking numbers of components coincide. (3) The corresponding knot types of components coincide. Slides: [YasutakaNakanishi.pdf]

 Hyo Won Park (KAIST) Minimal graphs whose braid groups are not a right angled Artin group For small braid indices we find minimal graphs whose braid groups are not a right angled Artin group. We show that their braid group are not a right angled Artin group by using such methods as cohomology algebras, Massey product, and combinatorial theory of commutator-relators groups. Slides: [HyoWonPark.pdf]

 Ruifeng Qiu (Dalian University of Technology) The Heegaard splittings of Haken 3-manifolds: a survey Let $M$ be a compact orientable 3-manifold, and $F$ be an incompressible surface in $M$ which cuts $M$ into two components $M_1$ and $M_2$. In this case, $M$ is called the surface sum of $M_1$ and $M_2$ along $F$. In this talk, we will introduce some recent results on the Heegaard splittings of surface sums. Slides: [RuifengQiu.pdf]

 Toshio Saito (Nara Women) On some classes of hyperbolic knots with the 3-sphere surgery For the distance of (1,1)-splittings of a knot in a closed orientable 3-manifold, it is an important problem whether a (1,1)-knot can admit (1,1)-splittings of different distance. In this paper, we give one-parameter families of hyperbolic (1,1)-knots such that each (1,1)-knot admits a Dehn surgery yielding the 3-sphere. It is remarkable that such knots are the first concrete examples each of whose (1,1)-splittings is of distance three. Slides: [ToshioSaito.pdf]

 Shin Satoh (Kobe University) Every non-trivial 2-knot needs four sheets The sheet number of a $2$-dimensional knot in $4$-space is an analogous quantity to the crossing number of a $1$-knot. For example, it is known that the spun trefoil and the $2$-twist-spun trefoil have the sheet number four. The aim of this talk is to prove that the sheet number of every non-trivial $2$-knot is greater than or equal to four. We also give another example of a $2$-knot with sheet number four whose Alexander polynomial is equal to $2t-1$. Slides: [ShinSatoh.pdf]

 Alexander Stoimenow (KAIST) Lie groups, Burau representation, and non-conjugate braids with the same closure link We use the unitarization of the Burau representation, found by Squier, to prove that if Squier's form is definite, the image of the representation is dense in the unitary group. We can also prove that if Budney's form unitarizing the Lawrence-Krammer representation is definite and the representation is irreducible, its image is dense in the unitary group. This implies that, except possibly for closures of full-twist braids, all links have infinitely many conjugacy classes of braid representations on any non-minimal number of (and at least 4) strands. Slides: [AlexanderStoimenow.pdf]

 Toshifumi Tanaka (Osaka City University) Knots and Casson handles We define the slice genus of a Casson handle CH as the minimal genus for a smooth connected orientable surface properly embedded in CH with boundary the attaching circle. In this talk, by using Rasmussen's s-invariant derived from Khovanov homology, we determine the slice genera for a certain infinite family of Casson handles. Slides: [ToshifumiTanaka.pdf]

 Shicheng Wang (Peking University) Some progress on Simon's conjecture. It is conjectured that each knot group surjects at most finitely many knot groups. We will report some progresses on this aspect. Slides: [ShichengWang.pdf]

 Yoshiro Yaguchi (Hiroshima University) Hurwitz orbits of braid systems of "standard" type, and its application Hurwitz action is viewed in the low demensional topology. Today, we see the Hurwitz orbits of braid systems of "standard" type. We also see its application to braided surfaces of "standard" type. Slides: [YoshiroYaguchi.pdf]

 Yuichi Yamada (University of Electro-Communications) Lens space surgery along ACampo divide knots II My recent research is "Some knots of exceptional Dehn surgeries are A'Campo's divide knots (coming from the singularity theory). How many and Why?" I will talk about difficulty and trial on Berge's Type8 family of lens space surgery. They are related to Rational homology 4-ball. Slides: [YuichiYamada.pdf]

 Zhiqing Yang (Dalian University of Technology) On embedding of infinite cyclic covering of knot space into three sphere We construct a type of knots whose commutator subgroups of the knot groups has $K^{\infty}$ (the free product of countable copies of some group $K\neq \{e\}$) as a quotient. Hence the infinite cyclic coverings can't be embedded into any compact three manifold. Slides: [ZhiqingYang.pdf]

 Bin Yu (Tongji University) Lorenz like Smale flows on 3-manifolds In this talk, we discuss how to realize Lorenz like Smale flows (LLSF) on 3-manifolds. It is an extension of M.Sullivan's work about Lorenz Smale flows on 3-sphere. LLSF is a special type of nonsingular Smale flow (NSF) i.e. it is a NSF with a repelling orbit, an attracting orbit, and a Lorenz like non-trivial saddle set. We focus on two questions: (1). Describe the topological conjugate classes of LLSF which can be realized on 3-sphere; (2). Which 3-manifolds admit LLSF? If it admits LLSF, how does it admit LLSF? The method we use is combinatorial. This topic can be looked as an attempt similarly with J. Morgan and M.Wada's work about Morse Smale flows on 3-manifolds. Slides: [BinYu.pdf]

 Mingxing Zhang (Dalian University of Technology) The Heegaard genera of surface sums (with Ruifeng Qiu and Shicheng Wang) Let $M$ be a compact orientable 3-manifold, and $F$ be an separating (resp. non-separating) incompressible surface in $M$ which cuts $M$ into two 3-manifolds $M_{1}$ and $M_{2}$ (resp. a manifold $M_{1}$). Then $M$ is called the surface sum (resp. self surface sum) of $M_1$ and $M_2$(resp. $M_1$) along $F$, denoted by $M=M_{1}\cup_{F} M_{2}$ (resp. $M=M_{1} \cup_{F}$). We will study how $g(M)$ is related to $\chi(F)$, $g(M_1)$ and $g(M_2)$ when both $M_{1}$ and $M_{2}$ have high distance Heegaard splittings. Slides: [MingxingZhang.pdf]