2018年度秋季総合分科会(於:岡山大学)

SUMMARY: This talk is concerned with the rational homotopy groups of the group \(\mathrm {Diff}(S^4)\) of self-diffeomorphisms of \(S^4\). We present a method to prove that there are many ‘exotic’ non-trivial elements in \(\pi _*\mathrm {Diff}(S^4)\otimes \mathbb {Q}\) parametrized by trivalent graphs. The proof utilizes Kontsevich’s characteristic classes for smooth sphere bundles and a version of clasper surgery for families, and is quite elementary. In fact, these are analogues of Chern–Simons perturbation theory in 3-dimension and clasper theory due to Goussarov and Habiro. We explain how the results in 3-dimension can be modified for 4-dimension and review some related problems, including the 4-dimensional Smale conjecture.

SUMMARY: Algebraic structures on the homology of free loop spaces have been studied under the name of string topology, the term first coined by Chas and Sullivan in their paper in 1999. Among various algebraic operations, the most fundamental is the loop product in homology. There are different requirements for a space \(X\) to admit a loop product on the homology of the free loop space \(LX\) over it. For example, Chas–Sullivan’s loop product is defined when \(X=M\) is a closed oriented manifold, while Chataur–Menichi’s loop product is for \(X=BG\) the classifying space of a group. We define a loop product when \(X=M_G\) is the Borel construction of a closed oriented manifold \(M\) acted by a Lie group \(G\). This provides a uniform treatment to the above two cases. We adopt a homotopy theoretic approach, which allows us to define a secondary version of the loop product and reveals an interesting connection to group homology. This is joint work with H. Tene.

SUMMARY: Let \(n\) be a positive integer. Dabkowski and Przytycki introduced the \(n\)th Burnside group of links which is preserved by \(n\)-moves, and proved that for any odd prime \(p\) there exist links which are not equivalent to trivial links up to \(p\)-moves by using their \(p\)th Burnside groups. This gives counterexamples for the Montesinos–Nakanishi \(3\)-move conjecture. In general, it is hard to distinguish \(p\)th Burnside groups of a given link and a trivial link. In this talk, we give a necessary condition for which \(p\)th Burnside groups are isomorphic to those of trivial links. The necessary condition gives us an efficient way to distinguish \(p\)th Burnside groups of a given link and a trivial link. As an application, we show that there exist links, each of which is not equivalent to a trivial link up to \(p\)-moves for any odd prime \(p\).

SUMMARY: For each positive integer \(n\) we introduce two local moves \(V(n)\) and \(V^n\), which are generalizations of the virtualization move. We give a classification of welded links up to \(V(n)\)-move. In particular, a \(V(n)\)-move is an unknotting operation on welded knots for any \(n\). On the other hand, we give a necessary condition for which two welded links are equivalent up to \(V^{n}\)-move. This leads to show that a \(V^{n}\)-move is not an unknotting operation on welded knots except \(n=1\). We also discuss relations among \(V^{n}\)-moves, associated core groups and the multiplexing of crossings.

SUMMARY: It is well known that any knot group has weight one, that is, it is normally generated by a single element. Such an element is called a weight element of the knot group. A meridian is a typical weight element, but it is known that the knot group of any non-trivial torus knot, hyperbolic \(2\)-bridge knot, cable knot, or hyperbolic knot with unknotting number one admits other weight elements. We show that for a few infinite classes of \(3\)-strand pretzel knots and all prime knots up to \(8\) crossings, the knot groups admit weight elements that are not automorphic images of meridians.

SUMMARY: Let \(K,K'\) be two-bridge knots of genus \(k,k'\) respectively. We show the necessary and sufficient condition of \(k\) in terms of \(k'\) that there exists an epimorphism from the knot group of \(K\) onto that of \(K'\).

SUMMARY: It is known that there is a fundamental inequality related to tunnel number of a knot and tangles obtained by its tangle decomposition. The inequality gives an upper bound for tunnel number of the knot, and there exists a knot in the 3-sphere so that the inequality is non-strict. This talk will discuss a slight generalization of those.

SUMMARY: We study \(2n\)-moves and the \(\Gamma \)-polynomial for knots. In this talk, we show that \(4k\)-move is not an unknotting operation for any integer \(k\geq 2\) by using the \(\Gamma \)-polynomial and if \(\Gamma (K;-1)=9\pmod {16}\) then the knot \(K\) does not become the unknot by a single \(4\)-move.

SUMMARY: Okada and the author showed that the sets of Alexander polynomials of knots obtained from \(5_1\) and \(10_{132}\) by a single crossing change does not coincide. In this talk, we give those of the granny knot and the square knot coincide.

SUMMARY: The interior polynomial is an invariant of (signed) bipartite graphs, and the interior polynomial of a plane bipartite graph is equal to a part of the HOMFLY polynomial of a naturally associated link. The HOMFLY polynomial \(P_L(v,z)\) is a famous link invariant with many known properties. For example, the HOMFLY polynomial of the mirror image of \(L\) is given by \(P_{L}(-v^{-1},z)\). This implies a property of the interior polynomial in the planar case. We prove that the same property holds for any bipartite graph. The proof relies on Ehrhart reciprocity applied to the so called root polytope. We also establish formulas for the interior polynomial inspired by the knot theoretical notions of flyping and mutation.

SUMMARY: We construct a 2-representation of the quiver Hecke (KLR) algebra of \(U_q(\mathfrak {gl}_m)\) on a bimodule category over a deformation of Webster algebra of type \(A_1\) categorifying the symmetric Howe representation of \(U_q(\mathfrak {gl}_m)\). As a consequence, we obtain a braid group action on the homotopy category of the bimodule category.

SUMMARY: We give the classification of the \(2g+1\)-dimensional complex linear representations of the pure mapping class groups of compact orientable surfaces of genus \(g\) with or without boundary/punctures, for sufficiently large \(g\). The classification is up to conjugation and is described in terms of certain twisted \(1\)-cohomology group of the pure mapping class group in question.

SUMMARY: A branched covering surface-knot is a surface-knot in the form of a branched covering over an oriented surface-knot, where we include the case when the number of branch points is zero. We can simplify a branched covering surface-knot by an addition of 1-handles with chart loops to a form such that its chart is the union of edges whose end points are vertices of degree 1, and 1-handles with chart loops. The simplifying number is the minimum number of 1-handles necessary to obtain such a simplified form. We give upper estimates of simplifying numbers for the case when branched covering surface-knots have a non-zero number of branch points.

SUMMARY: A branched twist spin is a 2-knot in the four sphere and it is a generalization of twist spun knot. In the study of four dimensional topology, the Gluck twist that is a surgery along a 2-knot is well-studied. It is known that the Gluck twist along a twist spun knot does not changed the four sphere. In this talk, we show the Gluck twist along a branched twist spin also does not changed the four sphere and construct a branched twist spin from another branched twist spin by Gluck twist along it.

SUMMARY: We give examples of Brieskorn homology spheres with \(E_8\)-genus \(g_8=1\), including \(\Sigma (2,5,9)\). This construction is a twisting method of well-known plumbing 4-manifolds. Our result contributes to Scaduto’s classification of intersection form of definite 4-manifolds which bound \(\Sigma (2,5,9)\).

SUMMARY: Let \(\mathfrak {g}\) be a semi-simple lie ring. We define quantum \(U_q(\mathfrak {g})\) invariant and perturbative \(\mathfrak {g}\) invariant of genus \(2\) handlebody-knots by an arithmetic expansion of quantum \(U_q(\mathfrak {sl}_{2})\) type invariant. It is effectively easy to calculate the perturbative invariant \(\mathfrak {g}=\mathfrak {sl}_{2}\), which is a significantly stronger invariant than the previous conventional invariant. A handlebody-knot is an embedding of a handlebody in the 3-sphere \(S^3\). Even for two types of handlebody knots, this can only be classified in up to six crossings. This information can be applied primarily to the 7-crossing-classification problem. Further, we aim to define universal perturbative invariants and introduce extensions upto such invariants.

SUMMARY: The Kauffman–Vogel polynomials are three variable polynomial invariants of \(4\)-valent rigid vertex graphs. A one-variable specialization of the Kauffman–Vogel polynomials for unoriented \(4\)-valent rigid vertex graphs was given by using the Kauffman bracket and the Jones–Wenzl idempotent with the color \(2\). Bataineh, Elhamdadi and Hajij generalized it to any color with even positive integers. We give another generalization of the one-variable Kauffman–Vogel polynomial for oriented and unoriented \(4\)-valent rigid vertex graphs by using the \(A_2\) bracket and the \(A_2\) clasps. These polynomial invariants are considered as the \(\mathfrak {sl}_3\) colored Jones polynomials for singular knots and links.

SUMMARY: We define a family of representations \(\{\rho _n\}_{n\geq 0}\) of a pure braid group \(P_{2k}\). These representations are obtained from an action of \(P_{2k}\) on a certain type of \(A_2\) web space with color \(n\). The \(A_2\) web space is a generalization of the Kauffman bracket skein module of a disk with marked points on its boundary. We also introduce a triangle-free basis of such an \(A_2\) web space and calculate matrix representations of \(\rho _n\) about the standard generators of \(P_{2k}\).

SUMMARY: We study the transition of the number of spirals in a Voronoi tiling for an Archimedean spiral lattice. The cut-and-project structure of the ‘grain boundaries’ is proved, by considering the continuous space of the parameters.

SUMMARY: In this talk, the notion of generic transversality and its characterization are given. The characterization is also a further improvement of the basic transversality result and its strengthening which was given by John Mather.

SUMMARY: In this talk, for a smooth map \(f_0\colon M \to \mathbb {R}^2\) of a compact surfaces with boundary into the plane, we determine the minimal number of the total of the number of cusps and the number of nodes among stable maps \(f\colon M \to \mathbb {R}^2\) which are homotopic to \(f_0\) by using some formula.

SUMMARY: We treat cuspidal edges with certain properties. We show relationships between signs of cusps appearing on Gauss maps of such cuspidal edges and geometric invariants of cuspidal edges.

SUMMARY: There exist basic sets for all zero-dimensional homeomorphisms: here the basic sets satisfy the following conditions: (1) they are closed sets, (2) an arbitrary orbit passes the basic sets at most once, (3) an arbitrary orbit passes every neighborhood of the basic set, (4) every periodic orbit passes them exactly once. For some cases of zero-dimensional homeomorphic systems, it seems possible to show the existence of non-atomic invariant measures. We propose another application. We used graph covering method when we constructed Bratteli–Vershik models. When we consider graphs whose edges have lengths, we can study classification of some portion of substitution subshifts.

SUMMARY: The question what is the condition of the substitution maps that generate minimal substitution subshifts is very basic. However, we could not find the answer to this question. We would like to report that we have found an answer to this question.

SUMMARY: We remark that there is no smooth function \(f(x)\) on \([0,1]\) which is flat at \(0\) such that the derivertive \(f^{(n)}\) of any order \(n \geq 0\) is positive on \((0,1]\). Moreover, the number of zeros of the \(n\)-th derivertive \(f^{(n)}\) grows to the infinity and the zeros accumulate to \(0\) when \(n \to \infty \).

SUMMARY: In this talk, I pick up some main points and talk my results for grading on spectra and notion of projective schemes. I will tell about definition of N-graded (resp. Z-graded) E-infinity-rings, by using an infinity-operad constructed from N (resp. Z) and spectral projective schemes.

SUMMARY: Milnor operation \(Q_m\) of Stiefel–Whitney class \(w_n\) is written explicitly as a certain determinant of \(2^m\)-dim matrix of \(w_n\)’s.

SUMMARY: In this talk we present the notion of smooth CW complexes and study their homotopy structures on the category of diffeological spaces. It is clear that the horn \(J^{n-1}=\partial I^{n-1} \times I \cup I^{n-1} \times \{1\}\) is a continuous retract of the \(n\)-cube \(I^n\). The fact contribute to the development of topological homotopy theory. But \(J^{n-1}\) is not a smooth retract of \(I^n\). Thus we introduce the notion of tame property such that \(J^{n-1}\) is an approximate retract of \(I^{n}\). We study smooth homotopy theory by using their properties.

SUMMARY: We construct a double chain complex generated by certain graphs and a chain map from that to the Chevalley–Eilenberg double complex of the dgl of symplectic derivations on a free dgl. It is known that the target of the map is related to characteristic classes of fibrations. We can describe some characteristic classes of fibrations whose fiber is a 1-punctured even-dimensional manifold by linear combinations of graphs though the cohomology of the dgl of derivations.

SUMMARY: We show a de Rham theory for cubical manifolds, and study rational homotopy type of the classifying spaces of smooth quandles. We also show that secondary characteristic classes produce cocycles of quandles.