## Tuesday Seminar on Topology

Seminar information archive ～06/09｜Next seminar｜Future seminars 06/10～

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 |

### 2007/11/20

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

A certain slice of the character variety of a knot group

and the knot contact homology

**長郷 文和**(東京工業大学大学院理工学研究科)A certain slice of the character variety of a knot group

and the knot contact homology

[ Abstract ]

For a knot $K$ in 3-sphere, we can consider representations of

the knot group $G_K$ into $SL(2,\\mathbb{C})$.

Their characters construct an algebraic set.

This is so-called the $SL(2,\\mathbb{C})$-character variety of

$G_K$ and denoted by $X(G_K)$.

In this talk, we introduce a slice (a subset) $S_0(K)$ of $X(G_K)$.

In fact, this slice is closely related to the A-polynomial

and the abelian knot contact homology.

For example, the A-polynomial $A_K(m,l)$ of a knot $K$ is

a two-variable polynomial knot invariant defined by using

the character variety $X(G_K)$.

Then we can show that for any {\\it small knot} $K$, the number of

irreducible components of $S_0(K)$ gives an upper bound of

the maximal degree of the A-polynomial $A_K(m,l)$ in terms of

the variable $l$.

Moreover, for any 2-bridge knot $K$, we can show that

the coordinate ring of $S_0(K)$ is exactly the degree 0

abelian knot contact homology $HC_0^{ab}(K)$.

We will mainly explain these facts.

For a knot $K$ in 3-sphere, we can consider representations of

the knot group $G_K$ into $SL(2,\\mathbb{C})$.

Their characters construct an algebraic set.

This is so-called the $SL(2,\\mathbb{C})$-character variety of

$G_K$ and denoted by $X(G_K)$.

In this talk, we introduce a slice (a subset) $S_0(K)$ of $X(G_K)$.

In fact, this slice is closely related to the A-polynomial

and the abelian knot contact homology.

For example, the A-polynomial $A_K(m,l)$ of a knot $K$ is

a two-variable polynomial knot invariant defined by using

the character variety $X(G_K)$.

Then we can show that for any {\\it small knot} $K$, the number of

irreducible components of $S_0(K)$ gives an upper bound of

the maximal degree of the A-polynomial $A_K(m,l)$ in terms of

the variable $l$.

Moreover, for any 2-bridge knot $K$, we can show that

the coordinate ring of $S_0(K)$ is exactly the degree 0

abelian knot contact homology $HC_0^{ab}(K)$.

We will mainly explain these facts.