Colored HOMFLY polynomials of knots presented as double fat diagrams

Abstract

Many knots and links in S 3 can be drawn as gluing of three manifolds with one or more four-punctured S 2 boundaries. We call these knot diagrams as double fat graphs whose invariants involve only the knowledge of the fusion and the braiding matrices of four -strand braids. Incorporating the properties of four-point conformal blocks in WZNW models, we conjecture colored HOMFLY polynomials for these double fat graphs where the color can be rectangular or non-rectangular representation. With the recent work of Gu-Jockers, the fusion matrices for the non-rectangular [21] representation, the first which involves multiplicity is known. We verify our conjecture by comparing with the [21] colored HOMFLY of many knots, obtained as closure of three braids. The conjectured form is computationally very effective leading to writing [21]-colored HOMFLY polynomials for many pretzel type knots and non-pretzel type knots. In particular, we find class of pretzel mutants which are distinguished and another class of mutants which cannot be distinguished by [21] representation. The difference between the [21]-colored HOMFLY of two mutants seems to have a general form, with A-dependence completely defined by the old conjecture due to Morton and Cromwell. In particular, we check it for an entire multi-parametric family of mutant knots evaluated using evolution method.

Factorization of differential expansion for antiparallel double-braid knots

“…However, despite a considerable progress, made in above references, nothing comparably impressive is yet achieved beyond torus knots -the problem turns to be extremely complicated. A new hope appeared with the introduction of the special class of double-fat knots in [94], where knot polynomials are presumably made from monodromy matrices of 4-point conformal blocks -and thus the CFT methods can be directly applied. This class is rather rich, in includes all the two-bridge and pretzel knots, moreover, it appeared to coincide with the arborescent knots, well known in mathematical literature [95,96].…”

“…Moreover, additional care is needed [36,37] in this case to formulate the arborescent calculus of [94] for multi-finger knots -the interaction vertices in the corresponding effective "field theory" are also not canonically defined (or "non-local"). Thus non-rectangular case will be further elaborated on elsewhere.…”

“…In section 6 we make a formal conjecture, that this property holds for arbitrary rectangular representations. Given all the recent achievements after [94] this basically reduces rectangular arborescent calculus to finding the DE coefficients in [105] for the rather simple one-parametric family of twist knots. For them a separate conjecture is needed, of which we cook up just a prototype in section 7: we make it in full generality for the single-floor pyramids.…”

“…Instead, the double evolution family of double-braid two-bridge knots definesS directly: in the notation of [94] H {m,n}…”

“…• Calculate H [r s ] for any desired arborescent knot by the technique of [36,37,94] (some additional care/checks can be needed with effective vertices in the multi-finger case) and make possible checks. The most artful part for today is the second step: revealing the structure of F for twist knots from very special examples -symmetric and antisymmetric representations R = [r] and R = [1 r ] and the only double-floor example, provided by R = [22].…”