2019年度秋季総合分科会(於:金沢大学)

SUMMARY: All oriented links \(L\) have special diagrams. Based on such a diagram we construct a sutured handlebody \(M\) which is closely related to the branched double cover of the link. From the sutured Floer homology of \(M\) we recover the Alexander polynomial \(\Delta \) of \(L\) via a simple forgetful map. More surprisingly, in cases when the diagram is also positive (so that \(L\) is a special alternating link), \(\mathrm {SFH}(M)\) can be used to compute those coefficients of the Homfly polynomial of \(L\) whose sum is the leading coefficient of \(\Delta \). To extract this information algebraically, we need the notion of the interior polynomial of a bipartite graph. Geometrically, this entails the cutting of some handles of \(M\) and identifying the resulting handlebody with a Seifert surface complement for another special alternating link. The talk involves joint results with A. Juhász, H. Murakami, A. Postnikov, J. Rasmussen, and D. Thurston.

SUMMARY: Let \(\Sigma \) be a compact connected oriented surface with nonempty boundary. We consider a homology cylinder \((C,c)\) of \(\Sigma \) where \(C\) is a compact connected \(3\)-manifold and \(c: \partial (\Sigma \times [0.1]) \to \partial C\) is a diffeomorphism such that \(c_1:\Sigma \to C,x \mapsto c(x,1)\) and \(c_0:\Sigma \to C, x \mapsto c(x,0)\) induce the same isomorphism in their homology groups. For \(i=0,1\), the embedding \(c_i\) induces the isomorphism \(c_{i*}:\widehat {\mathbb {Q} \pi _1} (\Sigma \times \{ i \} , (*,i) ) \to \widehat {\mathbb {Q} \pi _1} (C , (*,i))\) where \(* \in \partial \Sigma \). Here, for a manifold \(M\) and a base point \(P\), we denote \(\widehat {\mathbb {Q} \pi _1} (M,P) = {\lim }_{n \rightarrow \infty } \mathbb {Q} \pi _1 (M,P)/ (\ker \epsilon )^n\) where \(\epsilon \) is the augmentation map \( \mathbb {Q} \pi _1 (M,P) \to \mathbb {Q}, r \in \pi _1 (M,P) \mapsto 1\). For an embedding from a handlebody \(H_g\) of genus \(g\) into \(\Sigma \times [0,1]\) and an element \(\xi \) of the Torelli group of \(\partial H_g\), we denote by \(\Sigma \times [0.1]_{(e,\xi )}\) the \(3\)-manifold \(H_g\) and the closure of \(\Sigma \times [0,1] \backslash e(H_g)\) glued by \(e_{|\partial H_g} \circ \xi \). We remark the pair \((\Sigma \times [0,1]_{(e, \xi )}, \mathrm {id}_{\partial (\Sigma \times [0.1])})\) is a homology cylinder. In this talk, we obtain an invariant for homeomorphic classes of the set consisting of \(\Sigma \times [0,1]_{ (e, \xi )}\) for any embedding \(e\) of a handlebody and any element \(\xi \) of the Torelli group of the boundary of the handlebody using some skein algebra. This invariant depends only on the map \(c_{1*}{c_{0*}}^{-1}\). As an application, using this invariant, we give a method to compute the map \(c_{1*}{c_{0*}}^{-1}\).

SUMMARY: Two string links are equivalent up to \(2n\)-moves and link-homotopy if and only if their all Milnor link-homotopy invariants are congruent modulo \(n\). Moreover, the set of the equivalence classes forms a finite group generated by elements of order \(n\).

SUMMARY: The writhe polynomial is a fundamental invariant of an oriented virtual knot. It is known that two oriented virtual knots have the same writhe polynomial if and only if they are related by a finite sequence of shell moves. The aim of this talk is to classify oriented \(2\)-component virtual links up to shell moves by using several invariants of virtual links such as the linking numbers, \(n\)-writhes, and linking class.

SUMMARY: The link-homotopy classes of 4-component links were classified by Levine. We modify the result by using the clasper theory. This classification gives schematic and symmetric points of view to link-homotopy classes of 4-component links. We also give some new subsets of link-homotopy classes of 4-component links which are classified by invariants.

SUMMARY: Let \(A\) be a braided Hopf algebra \(A\) with braided commutativity. We introduce the space of \(A\) representations of a knot \(K\) by generalizing the \(G\) representation space of \(K\) defined for a group \(G\). By rebuilding the \(G\) representation space from the view point of Hopf algebras, it is extended to any braided Hopf algebra with braided commutativity. Applying this theory to \(\mathrm {BSL}(2)\) which is the braided quantum \(\mathrm {SL}(2)\) introduced by S. Majid, we get the space of \(\mathrm {BSL}(2)\) representations, which is a non-commutative algebraic scheme which provides quantized \(\mathrm {SL}(2)\) representations of \(K\). This is a joint work with Roland van der Veen.

SUMMARY: A combed 3-manifold is a pair of a closed oriented 3-manifold \(M\) and a homotopy class of non-singular vector fields on \(M\). Viewing combed 3-manifolds from virtual knot diagrams, we introduce an invariant of 3-manifolds.

SUMMARY: Satoh and Taniguchi introduced the \(n\)-writhe \(J_n\) for each non-zero integer \(n\), which is an invariant for virtual knots. They give a necessary and sufficient condition for a sequence of integers to be that of the \(n\)-writhes of a virtual knot. It is obvious that the virtualization of a real crossing is an unknotting operation for virtual knots. The unknotting number by a virtualization is called a virtual unknotting number and denoted by \(u^v(K)\). We have shown that for any given non-zero integer \(n\) and \(N\), there exists a virtual knot \(K\) with \(u^v(K)=1\) and \(J_n(K)=N\) in a previous paper. In this talk, we show that if \(\{c_n\}_{n \neq 0}\) is a sequence of integers with \(\sum _{n \neq 0} nc_n(K)=0\), then there exists a virtual knot \(K\) with \(u^v(K)=1\) and \(J_n(K)=c_n\) for any \(n \neq 0\). It is an extension of the previous result, and is a more powerful result.

SUMMARY: 4-move is a local change for knots which changes 4 half twists to 0 half twists or vice versa. In 1979, Yasutaka Nakanishi conjectured that 4-move is an unknotting operation. This is still an open problem. In this talk, we consider 4-move distance of knots, which is the minimal number of 4-moves needed to deform one into the other. In particular, the 4-move unknotting number of a knot is the 4-move distance to the trivial knot. We give a lower bound of the 4-move unknotting number and a table of the 4-move unknotting number of knots with up to 9 crossings. This is a joint work with Taizo Kanenobu.

SUMMARY: In this talk, Y. Nozaki, K. Sato and I introduce a new family of invariants of homology 3-spheres. These invariants are defined by using some filtered version of instanton Floer homology. Moreover, we show important properties of the invariants which give a family of subgroups of the homology cobordism group parametrized non-negative real numbers. In this point of view, we give a reproof of the result of Furuta and Fintushel–Stern and its generalization.

SUMMARY: Y. Nozaki, M. Taniguchi and the speaker introduced new homology cobordism invariants \(\{r_s\}_{s \in [-\infty , 0]}\) of homology 3-spheres. In particular, for any sequence of homology 3-spheres \(\{Y_n\}^{\infty }_{n=1}\), if (1) \(Y_1\) has a non-trivial \(r_s\), (2) \(-Y_1\) has trivial \(r_s\), and (3) there exists a simply connected negative definite cobordism with boundary \(Y_{n} \amalg -Y_{n+1}\) for each \(n\), then we can conclude the \(Y_n\) are linearly independent in the homology cobordism group. As an application, we give a sufficient condition for the linear independence of all positive \(1/n\)-surgeries on a knot in \(S^3\). As another application, we prove that the Whitehead doubles of all \((2,q)\)-torus knots with odd \(q \geq 3\) are linearly independent in the knot concordance group.

SUMMARY: K. Sato, M. Taniguchi and I constructed a homology cobordism invariant of integral homology 3-spheres to study the homology cobordism group. The invariant was computed only for some Brieskorn manifolds. In this talk, we compute the invariant of a certain hyperbolic 3-manifold and the resulting value seems to be irrational.

SUMMARY: We consider the Reidemeister torsion of a 3-manifold \(M\) for \(\mathit {SL}(2, \mathbf {C}\))-representations as a \(\mathbf {C}\)-valued function on the character variety of \(M\) and the image \(RT(M)\subset \mathbf {C}\) of this function. We prove that \(RT(M)\) is a finite set if \(M\) is the splice of two certain knots in \(S^3\). The proof is based on an observation on the character varieties and A-polynomials of knots. This is a joint work with Yuta Nozaki.

SUMMARY: A \(2\)-knot is a surface in \(\mathbf { R^4}\) that is homeomorphic to \(\mathbf {S^4}\), the standard sphere in \(3\)-space. A ribbon \(2\)-knot is a \(2\)-knot obtained from \(m\) \(2\)-spheres in \(\mathbf {R^4}\) by connecting them with \(m-1\) annuli. Let \(K^2\) be a ribbon 2-knot. The ribbon crossing number, denoted by \(r\)-\(cr(K^2)\) is a numerical invariant of the ribbon \(2\)-knot \(K^2\). We showed that there exist just \(17\) ribbon \(2\)-knots of the ribbon crossing number up to three. In this lecture we show that there exist no more than \(111\) ribbon \(2\)-knots of ribbon crossing number four.

SUMMARY: A branched covering surface in 4-space (a branched covering surface-knot) is a surface in 4-space in the form of a branched covering over a surface. For a branched covering surface, we have a numerical invariant called the simplifying number. We show that branched covering surfaces with degree three have the simplifying numbers less than three.

SUMMARY: We prove that a Legendrian knot \(K\) has two smoothly non-isotopic Lagrangian disks which fill \(K\). This implies that result of Ekholm that \(K\) has two Lagrangian disks which are non-Hamiltonian isotopic. The exteriors of two disks are diffeomorphic each other. This means that an included involution on the boundary (0-surgery on \(K\)) homeomorphically extends inside but cannot diffeomorphically extend inside.

SUMMARY: For deformation of 3-dimensional hyperbolic cone structures about cone angles \(\theta \), the local rigidity is known for \(0 \leq \theta \leq 2\pi \), but the global rigidity is known only for \(0 \leq \theta \leq \pi \). The proof of the global rigidity by Kojima is based on the fact that hyperbolic cone structures do not degenerate in deformation with decreasing cone angles. In this talk, we will construct hyperbolic cone structures on a link in \(T^{2} \times I\) explicitly. Then we will obtain an example of degeneration of hyperbolic cone structures with decreasing cone angles \(\pi < \theta < 2\pi \).

SUMMARY: Hitchin components are prefered connected components of \({\rm PSL}_n\mathbb {R}\)-character varieties of surface groups, which are higher dimensional analogs of Teichmüller spaces. By definition, they contain a subset corresponding to Teichmüller spaces, which is called the Fuchsian locus. In this talk, we show that the Fuchsian locus of Hitchin components corresponds to certain affine slice of the interior of a convex polytope under the Bonahon–Dreyer parameterization.

SUMMARY: We show a rigidity theorem for the Seiberg–Witten invariants mod 2 for families of spin 4-manifolds. We also give a family version of \(10/8\)-type inequality using this rigidity theorem. As an application, we shall prove the existence of a non-smoothable topological family of 4-manifolds whose fiber, base space, and total space are smoothable as manifolds. As its consequence, it follows that the inclusion map \({\rm Diff}(M)\hookrightarrow {\rm Homeo}(M)\) is not a weak homotopy equivalence for an oriented smooth 4-manifold \(M\) which is homeomorphic to \(K3\#n S^2\times S^2\) for \(n\geq 0\).

SUMMARY: Using finite dimensional approximations of families of Seiberg–Witten equations and Steenrod square operations, we shall give a new non-smoothable family of the \(K3\) surface. This implies the non-triviality of the fundamental group of the homotopy quotient of the homeomorphism group of \(K3\) divided by the diffeomorphism group of \(K3\).

SUMMARY: Let \(X\) be a path-connected separable metric space and \(D\) a countable dense subset of \(X\). For each \(d\in D\), let \(C_d\) be a circle and attach \(C_d\) to a point \(d\) so that the diameters of \(C_d\) converge to \(0\). We call the resulting space earring space \(E(X,D)\). If \(X\) is locally simply-connected and simply-connected, then \(\pi _(E(X,D))\) is a subgroup of the Hawaiian earring group. Since \(X\) is restored from \(\pi _1(E(X,D))\), we can investigate subgroups of the Hawaiian earring group using spaces \(X\).

SUMMARY: We introduce Markov-like functions on finite graphs and define the notation of the same pattern between those Markov-like functions. Then we show that two generalized inverse limits with Markov-like bonding functions on finite graphs having the same pattern are homeomorphic.

SUMMARY: In this talk, we shall characterize infinite-dimensional manifolds modeled on absorbing sets in non-separable Hilbert spaces by using the discrete cells property, which is a general position property based on their density.

SUMMARY: We calculate the mean dimension of full shifts over finite dimensional alphabets. We propose a problem which seems interesting from the viewpoint of infinite dimensional topology.

SUMMARY: According to K. T. Chen’s theory, we can obtain an isomorphisms between the Malcev Lie algebra of a manifold and a certain Lie algebra. We shall consider the space of such an isomorphims and its cohomology. The space can be regarded as a classifying space of a certain automorphism group of the fundamental group, and its cohomology gives characteristic classes of a fiber bundle.

SUMMARY: Diffeological spaces have been introduced by Souriau in the early 1980s. The notion generalizes that of a manifold. More precisely, the category \(\mathsf {Mfd}\) of finite dimensional manifolds embeds into \(\mathsf {Diff}\) the category of diffeological spaces, which is complete, cocomplete and cartesian closed. As an advantage, we can very naturally define a function space of manifolds in \(\mathsf {Diff}\) so that the evaluation map is smooth without arguments on infinite dimensional manifolds. We introduce a de Rham complex endowed with an integration map into the singular cochain complex which gives the de Rham theorem for every diffeological space.

SUMMARY: We give a spectral sequence converging to the space of knots in an oriented, simply connected closed manifold of dimension greater than 3. This spectral sequence has a computable E2-term. Construction of it is based on Goodwillie’s embedding calculus which approximates an embedding space of codimension greater than 2 by a homotopy limit of configuration spaces.

SUMMARY: In this talk, we will discuss a generating set of the rational loop homology algebra of the connected sum \({\mathbb C}P^{2}\# {\mathbb C}P^{2}\) and the rational loop product of simply-connected rationally elliptic closed 4-manifolds.

SUMMARY: Let \(N_{g,n}\) be a compact non-orientable surface of genus \(g\) with \(n\) boundary components. We give an infinite presentation for the subgroup of the mapping class group of \(N_{g,1}\) generated by all Dehn twists, for \(g\geq 3\).

SUMMARY: We consider two filtrations of the Torelli group: the lower central series and the Johnson filtration. We show the related graded Lie algebras are isomorphic up to degree 6.

SUMMARY: A diffeomorphism of surfaces is called hyperelliptic if it commutes with a hyperelliptic involution. Such diffeomorphisms naturally appear in the study of hyperelliptic curves. On the other hand, periodic diffeomorphisms play an important role in the study of mapping class groups of surfaces. Ishizaka classified up to conjugation hyperelliptic periodic diffeomorphisms of surfaces and gave Dehn twist presentations in terms of Humphries generators. In this talk, we will give an explicit decomposition of surfaces into pentagonal fundamental domains of hyperelliptic periodic diffeomorphisms. We apply it to obtain Dehn twist presentations which are different from those obtained by Ishizaka in general.

SUMMARY: A hyperbolic \(2\)-sphere is made from the double of an n-gon in Poincaré disc \((n \ge 3)\). Its geodesic flow is a transitive Anosov flow on a closed \(3\)-manifold. Birkhoff generalized a concept of a section for a flow. We call it Birkhoff section, that is an immersed surface with boundaries. We construct genus one Birkhoff sections for the geodesic flows of hyperbolic \(2\)-spheres with \(n\) singularities.

SUMMARY: We exhibit a relationship between the flux homomorphism on unit disk and the Euler class of flat \(\mathrm {Diff}_+(S^1)\)-bundle. We give a geometric construction of group cohomology class \(e_{\mathrm {Flux}}\) using the flux homomorphism and prove that \(e_{\mathrm {Flux}}\) is equal to the (universal) real Euler class of flat \(\mathrm {Diff}_+(S^1)\)-bundle up to constant multiple. Even as cocycles, we clarify a relation between them, which leads to the transgression formula connecting the flux homomorphism and the Euler cocycle.

SUMMARY: Reeb graphs of maps are fundamental tools in the theory of Morse functions, their higher dimensional versions and application to geometry of manifolds. A Reeb space is defined as the space of all connected components of inverse images of the map. In this talk, we consider the following fundamental and important problem: can we construct a smooth function satisfying several geometric conditions inducing a given graph as its Reeb graph? As a main result, we demonstrate construction of a smooth function satisfying several differential topological conditions on a 3-dimensional closed and orientable manifold inducing a given graph as the Reeb graph.

SUMMARY: In the industrial world, it is important to optimize several objectives such as cost, quality, safety and environmental impact. A multi-objective optimization problem is an optimization problem for such several objective functions. In this talk, we give a topological property of the set of optimal solutions of a strongly convex problem of class \(C^1\). Moreover, if we have time, then we also introduce an application of Singularity Theory to the problem. This talk is based on joint work mainly with Naoki Hamada.