On certain limits of the reduced colored Jones polynomials

Vortrag von Prof. Dr. Kazuo Habiro Datum: 13.04.10 Zeit: 15.00 - 17.00 Raum: Y27H46

Abstract:
The colored Jones polynomial is defined for a knot and a positive integer $n$
and takes values in the Laurent polynomial ring $Z[q,q^{-1}]$. Dasbach and
Lin studied the "head" and "tail" of the colored Jones polynomials. They
proved that, up to sign, the last and the first three coefficients of the
colored Jones polynomials of an alternating knot converge (i.e., are
independent of $n$ with finitely many exceptions), and conjectured that the
other coefficients also converge and yield two power series in $q$ and in
$q^{-1}$. Let us call these power series the head series and the tail series.
In this talk, I consider existence of the head and the tail series for the
reduced colored Jones polynomials, which are certain linear combinations of
the colored Jones polynomials, taking values in the Laurent polynomial ring.
We conjecture that the head and the tail series exist for the reduced colored
Jones polynomials for an alternating knot. Moreover, we conjecture that the
head (but not the tail) series exist for a positive knot and that all the
coefficients are nonnegative. Examples and partial results will also be given.