[English]
16:30 -- 18:00 Ȋwȓ(wLpX)
Tea: 16:00 -- 16:30 R[

Last updated January 18, 2006
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1011 -- 056, 16:30 -- 18:00

Vincent Blanloeil (IRMA, Univ. Louis Pasteur)

Cobordism of surfaces embedded in S4

Abstract: In this talk I will introduce cobordism of knots, and recall classical results of this theory. Then I will explain the results of a join work with O. Saeki, in which we gave a classification of embedded surfaces in S4 up to cobordism. As a consequence I will give a new proof of Rohlin's theorem on 3-manifolds embeddings in R5.

1018 -- 056, 16:30 -- 18:00

Andrei Pajitnov (Univ. de Nantes)

Novikov homology, twisted Alexander polynomials, and Thurston norm

Abstract: We discuss circle-valued Morse functions on complements to links in the 3-sphere. A generalization of Morse theory due to S.P.Novikov leads to computable numerical estimates of the number of critical points via the twisted Novikov homology, introduced in a paper of H.Goda and the author. We discuss a multi-variable analog of the twisted Novikov homology and its relations to the multi-variable twisted Alexander polynomials and the Thurston norm.

1025 -- 056, 16:30 -- 18:00

Daniel Matei (Institute of Mathematics of the Romanian Academy)

The local system homology of pure braids on surfaces

Abstract: Pure braid groups of surfaces, either closed or punctured, admit the structure of an iterated extension by free groups. This structure allows us to construct resolutions of the pure braid groups over the integers. These resolutions are then used to derive information on the rank one local system homology of the pure braids on surfaces.

111 -- 056, 16:30 -- 18:00

gi F (sw͌)

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1122 -- 056, 16:30 -- 18:00

؈ r (ww@Ȋw)

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1129 -- 056, 16:30 -- 18:00

˖{ B (cwHw)

The almost alternating diagrams of the trivial knot

Abstract: Bankwitz characterized an alternating diagram representing the trivial knot. A non-alternating diagram is called almost alternating if one crossing change makes the diagram alternating. We characterize an almost alternaing diagram representing the trivial knot. As a corollary we determine an unknotting number one alternating knot with a property that the unknotting operation can be done on its alternating diagram.

126 -- 056, 17:00 -- 18:00

R Fj (}g吔w)

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1213 -- 056, 16:30-- 18:00

On the best regularity for group actions on the interval

Given a continuous action of a finitely generated group on the interval, we will investigate what is the best regularity that can be achieved by performing topological conjugacies. As we will see, all the problem concentrates between the classes C1 and C2.

1220 -- 056, 16:30-- 18:30

Kq aK (ww@Ȋw)

Dehn surgery creating Klein bottles

We consider the situation that Dehn surgery on a knot in the $3$-sphere yields Klein bottles. Fix a knot $K \subset S^{3}$ and let $M$ be a complement. Assume that Dehn surgered manifolds $M(r)$ and $M(s)$ both contain Klein bottles. We discuss the geometric intersection number between such two slopes $r$ and $s$.

J (ww@Ȋw)

Chart description of monodromy representations on a closed surface.

Abstract: In surface-knot theory, S. Kamada introduced chart description to visualize a monodromy representation of a $2$-dimensional braid. We define a chart description for a monodromy representation on a closed surface valued in $G$. As an application, we describe chart move equivalences for Lefschetz fibrations and $(Z_2)^m\rtimes S_m$-monodromies.

110 -- 056, 16:30-- 18:30

c E (ww@Ȋw)

A nontrivial algebraic cycle in the Jacobian variety of the Klein quartic

Abstract: B. Harris defined the harmonic volume for a compact Riemann surface of genus $g\geq 3$, by means of Chen's iterated integrals. We prove some value of the harmonic volume for the Klein quartic $C$ is nonzero modulo $\frac{1}{2}\mathbb{Z}$, using special values of the generalized hypergeometric function. This result tells us the algebraic cycle $C-C^-$ is not algebraically equivalent to zero in the Jacobian variety $J(C)$.

ms (ww@Ȋw)

The primitive vector field and semi-analytic geometry of reflection groups

Abstract: The aim of this seminor is to investigate relations between geometry of a finite irreducible reflection group and that of its parabolic subgroups. Fundamental tools are the discriminant of reflection hyperplanes and the primitive vector field in the categorical quotient space of reflection group actions. Both of them play important roles in the theory of the flat (Frobenius) structure of reflection groups and construction of a semi-algebraic dual polyhedron, invented by K. Saito. I show that a connected component of common zero point set of algebroid functions defined by the discriminant is equivariant homeomorphic to a real vector space whose dimension is equal to the rank of a parabolic subgroup. In particular the case that a parabolic subgroup is of corank 1, this homeomorphism can be extended to semi-analytic diffeomorphism.

117 -- 056, 16:30 -- 18:00

Ryan Budney (Max Planck Institute)

Topology of spaces of knots

Abstract: I will describe what is known about the homotopy-type of the space of smooth embeddings of a j-sphere in an n-sphere, Emb(S^j,S^n), and a closely related space K_{n,j}, the space of "long embeddings" of R^j in R^n. I will try to survey the "group like" properties of these spaces: Haefliger's work on the isotopy classes of embeddings of S^j in S^n as monoids under the connected sum operation, and the iterated loop-space structures on certain "framed" analogues of K_{n,j}. There are results of Goodwillie and Weiss on the connectivity of K_{n,j} and Emb(S^j,S^n), and some computations of non-trivial homotopy groups of K_{n,1} for n>3 due to Sinha, Scannell, Longoni and Cattaneo. I will also give a description of the homotopy type of Emb(S^1,S^3) and K_{3,1}. The homotopy type of K_{3,1} is given recursively. Each component of K_{3,1} is described as the total space of a fiber bundle where the fiber is a product of "simpler" components of K_{3,1} and the base space can be one of various "elementary" spaces such as configuration spaces in R^2, and products of circles.

124 -- 056, 16:30-- 18:30

^ (ww@Ȋw)

Incompressible surfaces in 4-punctured sphere bundles

Abstract: Hatcher and Thurston classified the incompressible surfaces in the 2-bridge knot complements. And Floyd and Hatcher classified the incompressible surfaces in the 2-bridge link complements and in the punctured torus bundles over the circle. A natural extension of the technique used in these works is to apply for the classification of the incompressible surfaces in the 4-punctured sphere bundles over the circle. Indeed, Floyd and Hatcher mentioned this as a Remark in their paper. These works do not consider the connectedness of the surfaces. In this seminar, we will give a criterion for the connectedness and (non-)orientability of an incompressible surface of 2-bridge knot complement type in 4-punctured sphere bundles over the circle, and give the list of such surfaces with genus 0 or 1.

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