Abstract
I will talk about the Laplace transform of the HOMFLY-PT polynomial for knots, called the Harer-Zagier (HZ) transform, which is a function of two variables and q. For some special families of knots it admits a fully factorised form. This is not true, however, for the majority of knots, for which it can only be decomposed as a sum of factorised terms. An interesting relation between this decomposition and Khovanov homology will be discussed. Notwithstanding, we suggest that by xing the variable = qn, for some \magical” exponent n, the HZ transform of any knot can obtain a factorised form in terms of cyclotomic polynomials. Moreover, I will talk about the zeros of HZ transform which show an interesting behavior.
HOMFLZ polznomial and its Harer-Zagier transform.
The hyrer-Zagier Transform is like a geometric series in the Homfly polynomial. If I understand correctly this is a stronger version of the HOMFLY-polynomial.
Usually this function looks not very nive but Torus knots, Pretzel links and twisted torus knots have very nice descriptions.
Invariantrelationships
In polynomials with facotrized HZ the HOMFLY polynomial is related to the Kauffman polynomial. It also seems to be related to Khovanov homology.