KIAS Topology Seminar

Spatial graph theory is the study of graphs embedded in $S^3$. Most of work in this area has its roots in Conway and Gordon's results on intrinc properties. A graph is called intrinsically knotted if every embedding of the graph contains a non-trivially knotted graph.
In this talk, I will introduce the spatial graph theory, in particular intrinc knotting property and related results.

A finite simple graph is called thin-chordal if it has neither a square nor a path with 4 vertices as induced subgraphs. We show that a RAAG is thin-chordal if and only if its every finite index subgroup is a RAAG, and two RAAGs whose defining graphs are thin-chordal are qausi-isometric if and only if they are commensurable. Using the theorem, we classfies RAAGs in the class up to quasi-isometry (and commesurability).

In this talk, we will discuss Levi-Civita regularization and Moser regularization of the restricted three body problem and its limit problems. As a result of these regularizations, we will see how one can apply modern techniques in symplectic topology and contact topology to classical mechanics problems. We will define fiberwise convexity and fiberwise star-shapedness of Hamiltonian problems in Moser regularization and we will discuss their applications.

We compute integer valued knot concordance invariants of a family of general pretzel knots if the invariants are equal to the negative values of signatures for alternating knots. Examples of such invariants are Rasmussen $s$-invariants and twice Ozsváth-Szabó knot Floer homology $\tau$-invariants. We use the crossing change inequalities of Livingston and the fact that pretzel knots are almost alternating. As a consequence, for the family of pretzel knots given in this paper, $s$-invariants are twice $\tau$-invariants at the end.
In contrast, we also find an infinite family of 4-strand pretzel knots whose Rasmussen invariants are not equal to the negative values of signature invariants. In order to prove this, we use the long exact sequence of Khovanov homologies arisen from a link skein relation.

A lattice stick knot is a knot which consists of a finite number of straight line segments in the cubic lattice $Z^3$. The lattice stick number $s_L (K)$ of a knot $K$ is defined to be the minimal number of sticks required to construct the lattice stick knot. Minimum lattice length $Len(K)$ of a knot K is the minimum number of edges necessary to construct this lattice stick knot.
In this talk, we construct upper bounds of them in terms of its crossing number. And we find the exact values of them for small knots.

I plan to discuss the formulas for the edge coefficients of the Jones polynomial in semiqdequate diagrams and some applications to non-triviality of the Jones polynomial, odd crossing amphicheiral knots, etc.

The goal is to report on some long-term work on certain combinatorial properties of knot/link diagrams of given canonical genus. These turned out to have various ramifications and applications, including (1) enumeration of alternating knots by genus, (2) words in formal alphabets (Wicks forms), (3) graph embedding problems on surfaces, (4) markings and the $sl_N$ graph polynomial, (5) hyperbolic volume of polyhedra, graphs and links. I will try to explain (at least as far as time allows) some interrelations between these topics.

For a boundary-parabolic representation $\rho:\pi_1(K)\rightarrow PSL(2,C)$ of knot $K$, the developing map of $\rho$ can be constructed combinatorially using the shadow-colorings and it gives explicit volume formula. In this talk, we generalize this method to knotted trivalent graphs. Especially, this method determines the hyperbolic structures with parabolic meridians on the graph complement manifolds. Furthermore, the explicit volume formulas can be obtained combinatorially. This work is joint with Roland van der Veen of Leiden University.

A derivative of an algebraically slice knot $K$ is an oriented link disjointly embedded in a Seifert surface of $K$ such that its homology class forms a basis for a metabolizer of $K$. We use a ribbon obstruction related to Milnor's triple linking number. As an application we disprove the $n$-solvable filtration version of Kauffman's conjecture assuming that 0.5-solvable knot is also 1-solvable. In addition, for some algebraically slice knots with a fixed metabolizer, we get a complete understanding of Milnor's triple linking number of derivatives associated to the metabolizer. This is joint work with Mark Powell.

We introduce rational invariants called the slope invariants for all tunnels for tunnel number one knots in the $3$-sphere. The slope invariants arises from a study of the disk complex of the standard genus two handlebody in the $3$-sphere. The slope invariants have been calculated for some well known tunnels, including all tunnels for $2$-bridge knots and torus knots, and all $(1,1)$-tunnels. We introduce briefly a way to calculate them, and their several applications in other related topics.

Twisted torus knots are obtained by adding full twists to some parallel strands of torus knots. We will discuss some properties of these knots.

In the late 1990's, Cochran, Orr, and Teichner introduced the filtrations of the knot concordance group using gropes and Whitney towers in $4$-space. In this talk, we will discuss properties of these filtrations, and I will give new infinite rank subgroups of the successive quotients of the filtrations using amenable signatures and the notion of algebraic $n$-solutions.

In this talk we will give a brief overview of symplectic field theory (SFT). SFT is a framework for invariants of symplectic cobordisms, such as for example (smooth) affine varieties, coming from holomorphic curves. We will describe a little bit of the geometric and algebraic structure of this theory, and give some applications. We will give the necessary background on symplectic and contact geometry.

We constructed the first example of non-Kähler complex structures on $R^4$. Last November, I gave a talk about the construction at KIAS Complex Geometry and Topology seminar. This time we first recall our construction and show various properties of the complex manifold such as
(1) it cannot be holomorphically embedded in any compact complex surface,
(2) the Picard group is uncountable,
(3) the product with $C^{n-2}$ gives non-Kähler complex structures on $R^{2n}$ different from the Calabi-Eckmann structures.
This is a joint work with Antonio Jose Di Scala and Daniele Zuddas.

Joint with Dave Auckly, Paul Melvin, and Daniel Ruberman. It is known that the celebrated h-Cobordism theorem of Smale fails to hold for smooth 4-manifolds. The study of this failure led to the notion of a "cork", which is a contractible smooth submanifold $C$ in a closed 4-manifold $X$, with an involution $f$ on $\partial C$, such that removing $C$ from $X$ and regluing it by $f$ changes the diffeomorphism type of $X$. This operation is called "cork twist". The property that the cork twist diffeomorphism $f$ has order $2$ is interesting, but going back to 1990's, it has asked whether higher order corks may exist. That is a cork with a $n$-ordered diffeomorphism that generates $n$ different diffeomorphism types of $X$, possibly even infinite ordered diffeomorphism that gives all distinct diffeomorphism types. Recently, Tange constructed a cork with a $n$-ordered diffeomorphism, but that displays two diffeomorphism types. In this talk, I will discuss higher ordered corks, and more generally $G$-corks for any finite subgroup $G$ of $SO(4)$, where the cork twists corresponding to distinct elements of $G$ yield distinct diffeomorphism types on $X$ i.e. $|G|$-diffeomorphism types.

Cork is a pair $(C,t)$ of a contractible 4-manifold $C$ and a diffeomorphism $t$ of the boundary of $C$. The map $t$ extends to a homeomorphism but not diffeomorphism to inside $C$. Cork can be regarded as a localization of exotic 4-manifolds in some sense. I give examples of corks with any finite order $n$.

In this talk, I will discuss whether the set of tight fibered knots (including algebraic knots and $L$-space knots) is linearly independent in the knot concordance group. Note that, base on the work of Ken Baker, Tagami and I observed that if the slice-ribbon conjecture, then the set of tight fibered knots is linearly independent in the knot concordance group.

In a lens space $L(p, q)$ for a given homology class there exists a unique simple $(1, 1)$-knot yielding a homology sphere by integral surgery. First, we give families for Poincare sphere in detail. By computing such surgery data in general, we give some trend to the relation between surgery families.

Annulus twist is an operation on knots along an annulus embedded in the 3-sphere. I will survey several results on annulus twists. In particular, via annulus twists, I give a construction of slice knots (such that corresponding slice disks have the same exterior). Also, I explain a characterization of ribbon disks in terms of handle decompositions.

Let $(\mathcal{V},\mathcal{W};F)$ be a weakly reducible, unstabilized, genus three Heegaard splitting in an orientable, irreducible $3$-manifold $M$ and $\mathcal{D}_{\mathcal{VW}}(F)$ the subset of the disk complex $\mathcal{D}(F)$ consisting of simplices having at least one vertex from $\mathcal{V}$ and at least one vertex from $\mathcal{W}$. In this talk, we describe the shape of $\mathcal{D}_{\mathcal{VW}}(F)$, prove that there is a function from the components of $\mathcal{D}_{\mathcal{VW}}(F)$ to the isotopy classes of the generalized Heegaard splittings obtained by weak reductions from $(\mathcal{V},\mathcal{W};F)$, and observe the dynamics of $Mod(M,F)$ on $\mathcal{D}(F)$. In addition, we consider the way how to generalize these results for arbitrarily high genus cases.

In this talk, we discuss an odd-dimensional analogue of ABBV-localization theorem for circle actions. While the ABBV-localization technique can be applied for a circle action with non-empty fixed point set, our theorem can be applied for a fixed-point-free circle action. We also give an explicit formula of the Chern number associated to given pseudo-free circle action in terms of local data.