Department of Mathematics
OCAMI
Japanese
Friday Seminar on Knot Theory（2016）
Speaker：Celeste Damiani (Osaka City University, JSPS) 
Title：The many faces of Loop Braid Groups 
Date：February 10 (Fri.) 16：00～16：30 
Place：Dept. of Mathematics, Sci. Bldg., F415 

Loop braid groups, are a generalization of braid groups. These
groups have been an object of interest in different domains of mathematics
and mathematical physics, and have been called, in addition to loop braid
groups, with several names such as of motion groups, groups of
permutationconjugacy automorphisms, braidpermutation groups, welded braid
groups and untwisted ring groups. We unify all the formulations that have
appeared so far in the literature, with a complete proof of the equivalence
of these definitions. We also introduce an extension of these groups that
appears to be a more natural generalization of braid groups from the
topological point of view.
Speaker：Jieon Kim (Osaka City University, JSPS) 
Title：Marked graph diagrams of immersed surfacelinks 
Date：February 10 (Fri.) 16：30～17：00 
Place：Dept. of Mathematics, Sci. Bldg., F415 

An immersed surfacelink is the image of the disjoint union of
oriented surfaces in the 4space $\mathbb R^4$ by a smooth immersion. By
using normal forms of immersed surfacelinks defined by S. Kamada and K.
Kawamura, we define marked graph diagrams of immersed surfacelinks. In
addition, we generalize Yoshikawa moves for marked graph diagrams of
surfacelinks to local moves for marked graph diagrams of immersed
surfacelinks. We give some examples of marked graph diagrams of immersed
surfacelinks. This is a joint work with S. Kamada, A. Kawauchi and S. Lee.
Speaker： María de los Angeles Guevara Hernández（Instituto Potosino de Investigacion
Cientifica y Tecnologica, and Osaka City University） 
Title：Infinite families of prime knots with alternation number $1$ and
dealternating number $n$ 
Date：February 10 (Fri.) 17：00～17：30 
Place：Dept. of Mathematics, Sci. Bldg., F415 

The alternation number of a knot $K$, denoted by $alt(K)$, is the
minimum number of crossing changes necessary to transform a diagram of $K$
into some (possibly nonalternating) diagram of an alternating knot. And
the dealternating number of a knot $K$, denoted by $dalt(K)$, is the
minimum number of crossing changes necessary to transform a diagram $D$ of
$K$ into an alternating diagram. So, from these definitions it is immediate
that $alt(K) \leq dalt(K)$ for any knot $K$. In this talk, we will show
that for each positive integer $n$ there exists a family of infinitely
many hyperbolic prime knots with alternation number 1 and dealternating
number $n$.
Speaker：Atsushi Ishii (University of Tsukuba) 
Title：On augmented Alexander matrices 
Date：January 27 (Fri.) 16：00～17：00 
Place：Dept. of Mathematics, Sci. Bldg., F415 

We introduce augmented Alexander matrices, and construct link invariants.
An augmented Alexander matrix is defined with an Alexander triple, which gives
an extension of a quandle. This framework gives the twisted Alexander invariant
and the quandle cocycle invariant. This is a joint work with Kanako Oshiro.
Speaker：Naoki Sakata (Hiroshima University) 
Title：Veering triangulations of mapping tori of some pseudoAnosov maps arising from Penner's construction 
Date：January 13 (Fri.) 16：00～17：00 
Place：Dept. of Mathematics, Sci. Bldg., F415 

Agol proved that every pseudoAnosov mapping torus of a surface, punctured along the singular points of the stable and unstable foliations, admits a canonical "veering" ideal triangulation.
In this talk, I will describe the veering triangulations of the mapping tori of some pseudoAnosov maps arising from Penner's construction.
Speaker  ：  Takuya Ukida (Tokyo Institute of Technology) 
Title  ：  Planar Lefschetz fibrations and Stein structures with distinct OzsvathSzabo
invariants on corks 
Date  ：  December 9 (Fri.) 16：00～17：00 
Place  ：  Dept. of Mathematics, Sci. Bldg., F415 

Thanks to a result of Lisca and Matic and a refinement by Plamenevskaya,
it is known that on a 4manifold with boundary Stein structures with nonisomorphic
$Spin^c$ structures induce contact structures with distinct OzsvathSzabo
invariants. Here we give an infinite family of examples showing that converse
of LiscaMaticPlamenevskaya theorem does not hold in general. Our examples
arise from Mazur type corks.
Speaker  ：  Shin Satoh (Kobe University, Graduate School of Science, Department of
Mathematics) 
Title  ：  The ribbon stable class of a surfacelink 
Date  ：  December 2 (Fri.) 16：00～17：00 
Place  ：  Dept. of Mathematics, Sci. Bldg., F415 

Two orientable surfacelinks are called stably equivalent if they are ambient
isotopic in $4$space up to adding or deleting trivial $1$handles. In
this talk, we construct a map $\omega$ from the set of orientable surfacelinks
to that of stable equivalence classes of ribbon surfacelinks, and study
several properties of $\omega$. In particular, we prove that $\omega(F)=[F]$
for any ribbon surfacelink $F$, that $F$ and $\omega(F)$ have the same
fundamental quandle, that $\omega(F)$ has a representative of genus $1$
for any deformspinning of a $2$bridge knot, and that $\omega(\tau^0 L)=\omega(\tau^1
L)$ for any link $L$, where $\tau^k L$ is the $k$turned spinning of $L$.
Speaker  ：  Takefumi Nosaka (Kyushu University, Faculty of Mathematics) 
Title  ：  Milnor invariants via unipotent Magnus embeddings 
Date  ：  November 18 (Fri.) 16：00～17：00 
Place  ：  Dept. of Mathematics, Sci. Bldg., F415 

We reconfigure Milnor invariant, in terms of central group extensions
and unipotent Magnus embeddings, and develop a diagrammatic computation
of the invariant. In this talk, we explain the reconfiguration and
the computation with mentioning some examples. This is a joint work
with Hisatoshi Kodani.
Speaker  ：  Tetsuya Ito (Osaka University) 
Title  ：  Biordering and Alexander invariants 
Date  ：  November 4 (Fri.) 16：00～17：00 
Place  ：  Dept. of Mathematics, Sci. Bldg., F415 

We show that if a knot group (more generally, finitely presented group)
is biorderable then its Alexander polynomial has at least one positive
real root.
Speaker  ：  Katsumi Ishikawa (Research Institute for Mathematical Sciences, Kyoto University) 
Title  ：  On the classification of smooth quandles 
Date  ：  October 28 (Fri.) 16：00～17：00 
Place  ：  Dept. of Mathematics, Sci. Bldg., F415 

A smooth quandle is a differentiable manifold with a smooth quandle
operation.
We show that every smooth transitive connected quandle is
isomorphic to a homogeneous space with an operation defined from a group
automorphism.
We also give an explicit classification of such quandles of dimension 1 and
2.
Speaker  ：  Genki Omori (Tokyo Institute of Technology) 
Title  ：  A small normal generating set for the handlebody subgroup of the
Torelli group 
Date  ：  October 21 (Fri.) 15：00～16：00 
Place  ：  Dept. of Mathematics, Sci. Bldg., F415 

We consider the handlebody subgroup of the Torelli group, i.e. the
intersection of the handlebody group and the Torelli group of an orientable
surface. The handlebody subgroup of the Torelli group is related to
integral homology 3spheres through the Heegaard splittings. In this talk,
we give a small normal generating set for the handlebody subgroup of the
Torelli group.
Speaker  ：  Teruhisa Kadokami (Kanazawa University) 
Title  ：  Three amphicheiralities of a virtual link 
Date  ：  july 22 (Fri.) 16：00～17：00 
Place  ：  Dept. of Mathematics, Sci. Bldg., F415 

We define three amphicheiralities for a virtual link by
using its geometric realization.
Speaker  ：  Yuta Nozaki (The University of Tokyo) 
Title  ：  An explicit relation between knot groups in lens spaces and those in $S^3$ 
Date  ：  July 15 (Fri.) 16：00～17：00 
Place  ：  Dept. of Mathematics, Sci. Bldg., F415 

We consider a $p$fold cyclic covering map $(S^3, K) \to
(L(p,q), K')$ and describe the knot group $\pi_1(S^3 \setminus K)$ in
terms of $\pi_1(L(p,q) \setminus K')$. As a consequence, we give an
alternative proof for the fact that a certain knot in $S^3$ cannot be
represented as the preimage of any knot in a lens space. In the proof,
the subgroup of a group $G$ generated by the commutators and the $p$th
power of each element of $G$ plays a key role.
Speaker  ：  Akio Kawauchi (OCAMI) 
Title  ：  On a crosssection of an immersed spherelink in 4space 
Date  ：  April 8 (Fri.) 16：00～17：00 
Place  ：  Dept. of Mathematics, Sci. Bldg., F415 

The torsion Alexander polynomial, the reduced
torsion Alexander polynomial and the local signature
invariant of a crosssection of an immersed spherelink are investigated from the viewpoint of how to influence to the immersed spherelink. It is shown that the torsion Alexander polynomial of a symmetric middle crosssection of a ribbon spherelink is an invariant of the ribbon spherelink. A generalization to a symmetric middle
crosssection of an immersed ribbon spherelink is given.
Speaker  ：  Sang Youl Lee (Pusan National University) 
Title  ：  The quantum $A_2$ polynomial for oriented virtual links 
Date  ：  May 13 (Fri.) 16：00～17：00 
Place  ：  Dept. of Mathematics, Sci. Bldg., F415 

A tangled trivalent graph diagram is an oriented link diagram possibly with some trivalent vertices whose incident edges are oriented all inward or all outward.
Two tangled trivalent graph diagrams are said to be regular isotopic if
they are transformed into each other by a finite sequence of classical Reidemeister moves of type 2, type 3 and trivalent vertex passing moves.
In 1994, G. Kuperberg derived an inductive, combinatorial definition of a polynomialvalued regular isotopy invariant (called the quantum $A_2$ invariant) of
links and tangled trivalent graph diagrams with values in the integral Laurent polynomial ring $\mathbb Z[a, a^{1}]$ that equals the ReshetikhinTuraev invariant corresponding
to the simple Lie algebra $A_2$.
In this talk, I would like to talk about an extension of the quantum $A_2$ invariant to virtual tangled trivalent graph diagrams and a derived polynomial invariant
for oriented virtual links satisfying a certain skein relation.
Last Modified on February 7, 2017.
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