We develop a diagrammatic calculus for representations of unrolled quantum sl2 at a fourth root of unity. This allows us to prove Seifert-Torres type formulas for certain splice links using quantum ...algebraic methods, rather than topological methods. Other applications of this diagrammatic calculus given here are a skein relation for n-cabled double crossings and a simple proof that the quantum invariant associated with these representations determines the multivariable Alexander polynomial.
We rewrite the recently proposed differential expansion formula for HOMFLY polynomials of the knot 4
1
in an arbitrary rectangular representation R = r
s
as a sum over all Young subdiagrams λ of R ...with surprisingly simple coefficients of the Z factors. Intriguingly, these coefficients are constructed from the quantum dimensions of symmetric representations of the groups SL(r) and SL(s) and restrict the summation to diagrams with no more than s rows and r columns. Moreover, the β-deformation to Macdonald dimensions yields polynomials with positive integer coefficients, which are plausible candidates for the role of superpolynomials for rectangular representations. Both the polynomiality and the positivity of the coefficients are nonobvious, nevertheless true. This generalizes the previously known formulas for symmetric representations to arbitrary rectangular representations. The differential expansion allows introducing additional gradings. For the trefoil knot 3
1
, to which our results for the knot 4
1
are immediately extended, we obtain the so-called fourth grading of hyperpolynomials. The property of factorization in roots of unity is preserved even in the five-graded case.
The recently proposed Kameyama–Nawata–Tao–Zhang (KNTZ ) trick completed the long search for exclusive Racah matrices
and
for all rectangular representations. The success of this description is a ...remarkable achievement of modern knot theory and classical representation theory, which was initially considered a tool for knot calculus but instead turned out to be its direct beneficiary. We show that this approach in fact consists in converting the arborescent evolution matrix
into the triangular form
, and we demonstrate how this works and show how the previous puzzles and miracles of the differential expansions look from this standpoint. Our conjecture for the form of the triangular matrix
in the case of the nonrectangular representation
is completely new. No calculations are simplified in this case, but we explain how it all works and what remains to be done to completely prove the conjecture. The discussion can also be useful for extending the method to nonrectangular cases and for the related search for gauge-invariant arborescent vertices. As one more application, we present a puzzling, but experimentally supported, conjecture that the form of the differential expansion for all knots is completely described by a particular case of twist knots.
Are there p-adic knot invariants? Morozov, A. Yu
Theoretical and mathematical physics,
04/2016, Letnik:
187, Številka:
1
Journal Article
Recenzirano
Odprti dostop
We suggest using the Hall–Littlewood version of the Rosso–Jones formula to define the germs of p-adic HOMFLY-PT polynomials for torus knots
m
,
n
as coefficients of superpolynomials in a ...q-expansion. In this form, they have at least the
m
,
n
↔
n
,
m
topological invariance. This opens a new possibility to interpret superpolynomials as p-adic deformations of HOMFLY polynomials and poses a question of generalizing to other knot families, which is a substantial problem for several branches of modern theory.
In this paper, we present a new algorithm to evaluate the Kauffman bracket polynomial. The algorithm uses cyclic permutations to count the number of states obtained by the application of ‘
A’ and ‘
...B’ type smoothings to the each crossing of the knot. We show that our algorithm can be implemented easily by computer programming.
The classification of protein structures is an important and still outstanding problem. The purpose of this paper is threefold. First, we utilize a relation between the Tutte and homfly polynomial to ...show that the Alexander-Conway polynomial can be algorithmically computed for a given planar graph. Second, as special cases of planar graphs, we use polymer graphs of protein structures. More precisely, we use three building blocks of the three-dimensional protein structure--alpha-helix, antiparallel beta-sheet, and parallel beta-sheet--and calculate, for their corresponding polymer graphs, the Tutte polynomials analytically by providing recurrence equations for all three secondary structure elements. Third, we present numerical results comparing the results from our analytical calculations with the numerical results of our algorithm-not only to test consistency, but also to demonstrate that all assigned polynomials are unique labels of the secondary structure elements. This paves the way for an automatic classification of protein structures.
This thesis examines graph polynomials and particularly their complexity. We give short proofs of two results from Gessel and Sagan (1996) which present new evaluations of the Tutte polynomial ...concerning orientations. A theorem of Massey et al (1997) gives an expression concerning the average size of a forest in a graph. We generalise this result to any simplicial complex. We answer a question posed by Kleinschmidt and Onn (1995) by showing that the language of partitionable simplicial complexes is in NP. We prove the following result concerning the complexity of the Tutte polynomial: Theorem 1. For any fixed k, there exists a polynomial time algorithm A, which will input any graph G, with tree-width at most k, and rational numbers x and y, and evaluate the Tutte polynomial, T(G;x,y). The rank generating function S of a graphic 2-polymatroid was introduced by Oxley and Whittle (1993). It has many similarities to the Tutte polynomial and we prove the following results. Theorem 2. Evaluating S at a fixed point (u,v) is #P-hard unless uv=1 when there is a polynomial time algorithm. Theorem 3. For any fixed k, there exists a polynomial time algorithm A, which will input any graph G, with tree-width at most k, and rational numbers u and v, and evaluate S(G;u,v). We consider a class of graphs $S$, which are those graphs which are obtainable from a graph with no edges using the unsigned version of Reidemeister moves. We examine the relationship between this class and other similarly defined classes such as the delta-wye graphs. There remain many open questions such as whether S contains every graph. However we have an invariant of the moves, based on the Tutte polynomial, which allows us to determine from which graph with no edges, if any, a particular graph can be obtained. Finally we consider a new polynomial on weighted graphs which is motivated by the study of weight systems on chord diagrams. We give three states model and a recipe theorem. An unweighted version of this polynomial is shown to contain as specialisations, a wide range of graph invariants, such as the Tutte polynomial, the polymatroid polynomial of Oxley and Whittle (1993) and the symmetric function generalisation of the chromatic polynomial introduced by Stanley (1995). We close with a discussion of complexity issues proving hardness results for very restricted classes of graphs.
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