Harer-Zagier formulas for knot matrix models

Abstract

Knot matrix models are defined so that the averages of characters are equal to knot polynomials. From this definition one can extract single trace averages and generation functions for them in the group rank -which generalize the celebrated Harer-Zagier formulas for Hermitian matrix model. We describe the outcome of this program for HOMFLY-PT polynomials of various knots. In particular, we claim that the Harer-Zagier formulas for torus knots factorize nicely, but this does not happen for other knots. This fact is mysteriously parallel to existence of explicit β = 1 eigenvalue model construction for torus knots only, and can be responsible for problems with construction of a similar model for other knots.

Intersecting Defects and Supergroup Gauge Theory

“…9 Therefore, having in mind the matrix model description of the unknot in Chern-Simons theory (e.g. [82,83] and references therein) or its refined version [28,29], we have the identification…”

Punctures and p-spin curves from matrix models III. Dl type and logarithmic potential

“…The knot matrix model is discussed through Harer-Zagier (HZ) formula defined below [50,51], which becomes a generating function of Gaussian mean. The knot polynomial of HOMFLYPT for a torus knot is realized as an average of a character [51].…”

“…The knot matrix model is discussed through Harer-Zagier (HZ) formula defined below [50,51], which becomes a generating function of Gaussian mean. The knot polynomial of HOMFLYPT for a torus knot is realized as an average of a character [51]. For instance, the trefoil T (2, 3), which is a torus knot of (2,3) type, and the normalized HOMFLYPT polynomial P , by the multiplication of (v − v −1 )/z with the replacement of v → q N and z → q − q −1 , is expressed as P (T (2, 3) : v, z) = (q 2 + q −2 )q 5N − (q 2 + 1 + q −2 )q 3N + q N q − q −1 (9.5)…”

“…For the strong coupling expansion of p < 0, the relation to knot polynomials and Laplace-Borel transform is discussed in this article. The non-torus knots has no factorization [51] and its representation by a matrix model will be an interesting subject in a future.…”

“…The n point function U (s 1 , ..., s n ) is interpreted as the Wilson loops, which are generating functions of the following Gaussian means< trM i1 • • • trM in >= trM i1 • • • trM in e − 1 2 trM 2 dM e − 1 2 trM 2 dM (9.1)where M is N × N Hermitian matrix. HZ function is defined by[51,52] N − 1] (9.2)from which one can obtain, for instance,< trM 4 >= 2N 3 + N (9.3)By the sum of N , and by the use of HZ transform, one can write a Laplace transform of above quantity,Z(λ) = ∞ N =0 λ N < trM 4 >= ∞ N =0 (2N 3 + N )λ N = 3λ(1 + λ) (1 − λ) 4 . (9.4) …”