## Tuesday Seminar on Topology

Seminar information archive ～08/08｜Next seminar｜Future seminars 08/09～

Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |

### 2006/12/19

16:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Poisson structures on the homology of the spaces of knots

On projections of pseudo-ribbon sphere-links

**境 圭一**(東京大学大学院数理科学研究科) 16:30-17:30Poisson structures on the homology of the spaces of knots

[ Abstract ]

We study the homological properties of the space $K$ of (framed) long knots in $\\R^n$, $n>3$, in particular its Poisson algebra structures.

We had known two kinds of Poisson structures, both of which are based on the action of little disks operad. One definition is via the action on the space $K$. Another comes from the action of chains of little disks on the Hochschild complex of an operad, which appears as $E^1$-term of certain spectral sequence converging to $H_* (K)$. The main result is that these two Poisson structures are the same.

We compute the first non-trivial example of the Poisson bracket. We show that this gives a first example of the homology class of $K$ which does not directly correspond to any chord diagrams.

We study the homological properties of the space $K$ of (framed) long knots in $\\R^n$, $n>3$, in particular its Poisson algebra structures.

We had known two kinds of Poisson structures, both of which are based on the action of little disks operad. One definition is via the action on the space $K$. Another comes from the action of chains of little disks on the Hochschild complex of an operad, which appears as $E^1$-term of certain spectral sequence converging to $H_* (K)$. The main result is that these two Poisson structures are the same.

We compute the first non-trivial example of the Poisson bracket. We show that this gives a first example of the homology class of $K$ which does not directly correspond to any chord diagrams.

**吉田 享平**(東京大学大学院数理科学研究科) 17:30-18:30On projections of pseudo-ribbon sphere-links

[ Abstract ]

Suppose $F$ is an embedded closed surface in $R^4$.

We call $F$ a pseudo-ribbon surface link

if its projection is an immersion of $F$ into $R^3$

whose self-intersection set $\\Gamma(F)$ consists of disjointly embedded circles.

H. Aiso classified pseudo-ribbon sphere-knots ($F$ is a sphere.)

when $\\Gamma(F)$ consists of less than 6 circles.

We classify pseudo-ribbon sphere-links

when $F$ is two spheres and $\\Gamma(F)$ consists of less than 7 circles.

Suppose $F$ is an embedded closed surface in $R^4$.

We call $F$ a pseudo-ribbon surface link

if its projection is an immersion of $F$ into $R^3$

whose self-intersection set $\\Gamma(F)$ consists of disjointly embedded circles.

H. Aiso classified pseudo-ribbon sphere-knots ($F$ is a sphere.)

when $\\Gamma(F)$ consists of less than 6 circles.

We classify pseudo-ribbon sphere-links

when $F$ is two spheres and $\\Gamma(F)$ consists of less than 7 circles.