Department of Mathematics OCAMI Japanese

Friday Seminar on Knot Theory（2016）

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 permutation-conjugacy automorphisms, braid-permutation 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.

An immersed surface-link is the image of the disjoint union of oriented surfaces in the 4-space $\mathbb R^4$ by a smooth immersion. By using normal forms of immersed surface-links defined by S. Kamada and K. Kawamura, we define marked graph diagrams of immersed surface-links. In addition, we generalize Yoshikawa moves for marked graph diagrams of surface-links to local moves for marked graph diagrams of immersed surface-links. We give some examples of marked graph diagrams of immersed surface-links. This is a joint work with S. Kamada, A. Kawauchi and S. Lee.

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 non-alternating) 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 $n$.

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.

Agol proved that every pseudo-Anosov 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 pseudo-Anosov maps arising from Penner's construction.

Thanks to a result of Lisca and Matic and a refinement by Plamenevskaya, it is known that on a 4-manifold with boundary Stein structures with non-isomorphic $Spin^c$ structures induce contact structures with distinct Ozsvath-Szabo invariants. Here we give an infinite family of examples showing that converse of Lisca-Matic-Plamenevskaya theorem does not hold in general. Our examples arise from Mazur type corks.

Two orientable surface-links 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 surface-links to that of stable equivalence classes of ribbon surface-links, and study several properties of $\omega$. In particular, we prove that $\omega(F)=[F]$ for any ribbon surface-link $F$, that $F$ and $\omega(F)$ have the same fundamental quandle, that $\omega(F)$ has a representative of genus $1$ for any deform-spinning 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 $L$.

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.

We show that if a knot group (more generally, finitely presented group) is bi-orderable then its Alexander polynomial has at least one positive real root.

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.

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 3-spheres through the Heegaard splittings. In this talk, we give a small normal generating set for the handlebody subgroup of the Torelli group.

We define three amphicheiralities for a virtual link by using its geometric realization.

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.

The torsion Alexander polynomial, the reduced torsion Alexander polynomial and the local signature invariant of a cross-section of an immersed sphere-link are investigated from the viewpoint of how to influence to the immersed sphere-link. It is shown that the torsion Alexander polynomial of a symmetric middle cross-section of a ribbon sphere-link is an invariant of the ribbon sphere-link. A generalization to a symmetric middle cross-section of an immersed ribbon sphere-link is given.

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 polynomial-valued 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 Reshetikhin-Turaev 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.