On universal knot polynomials

Abstract

We present a universal knot polynomials for 2-and 3-strand torus knots in adjoint representation, by universalization of appropriate Rosso-Jones formula. According to universality, these polynomials coincide with adjoined colored HOMFLY and Kauffman polynomials at SL and SO/Sp lines on Vogel's plane, respectively and give their exceptional group's counterparts on exceptional line. We demonstrate that [m,n]=[n,m] topological invariance, when applicable, take place on the entire Vogel's plane. We also suggest the universal form of invariant of figure eight knot in adjoint representation, and suggest existence of such universalization for any knot in adjoint and its descendant representations. Properties of universal polynomials and applications of these results are discussed.

Checks of integrality properties in topological strings

“…The progress is a cumulative effect of merging of the different research directions: i) reformulation of the RT formalism in the spaces of intertwining operators [73][74][75][76]- [80] with developments of the highest weight technique [31,33] and the eigenvalue conjecture [1,78] to evaluate the inclusive and exclusive Racah matrices [33]- [35], ii) representing the knot polynomials for all arborescent knots through the exclusive Racah matrices S andS [29], iii) developing the family technique [30,32,144,145] in order to adequately classify knots, at least, for calculational purposes, iv) applying Vogel's universality [154] to handle the adjoint representations and their descendants; important here is that deviations from the universality at the group theory level are not seen in knot polynomial calculus [1,141].…”

“…The hypothesis actually originated from knot theory studies, and the idea was to raise it up to the group theory level, where it partly failed. However, not very surprisingly, the knot polynomials are not sensitive to the failures, and they are indeed universal [1,141]. Moreover, an extension of Vogel's hypothesis from the dimensions and Casimirs to the Racah matrices, which is one of the steps required for evaluating the adjoint HOMFLY polynomial, also provided the non-trivial confirmation of the eigenvalue hypothesis and explicit formulas for the 6 × 6 Racah matrices [1].…”

“…Finding these Racah matrices for arbitrary representation gets especially difficult in the case of non-trivial multiplicities (i.e. for non-rectangular Young diagrams and outside the E 8 -sector [141]). Such representations with non-trivial multiplicity plays a crucial role in distinguishing mutant knot pairs [29]- [35].…”

Eigenvalue conjecture and colored Alexander polynomials

“…The situation with the eigenvalue conjecture is much worse: it was quite difficult to check it for the matrix sizes (dimensions of W (Q)) 2,3,4,5 even in the simplest case of m = 3 strands. For size 6 it was validated very recently within the framework of the knot universality of [54,55] and by application to advanced Racah calculus in [46,48,49]. This was an important step, because, beginning from the size 6, the eigenvalue conjecture does not immediately follow from the Yang-Baxter relations only [18,56]; still for knot calculus it works well.…”

HOMFLY polynomials in representation [3, 1] for 3-strand braids

“…For the particular case of sl 4 algebra, [2,1,1] is the adjoint representation and the adjoint HOMFLY polynomial satisfies [101,107] the universality hypothesis [108], unifying them with the adjoint polynomials for other groups, including the much simpler adjoint Kauffman polynomials (they are simpler because the adjoint representation of so N is just [1,1] for all N , while it is an N -dependent [2, q N −2 ] for sl N ). This allows one to compare our H [3,1] at A = q 4 with the universal formulas from [101].…”