Abstract. In this paper, we study the quantum sl(n) representation category using the web space. Specially, we extend sl(n) web space for n ≥ 4 as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial Pn(q) specialized to a one variable polynomial can be computed by a linear expansion with respect to a presentation of the quantum representation category of sl(n). Moreover, we correct the false conjecture [30] given by Chbili, which addresses the relation between some link polynomials of a periodic link and its factor link such as Alexander polynomial (n = 0) and Jones polynomial (n = 2) and prove the corrected conjecture not only for HOMFLY polynomial but also for the colored HOMFLY polynomial specialized to a one variable polynomial.
In 2013, Cheng and Gao introduced the writhe polynomial of virtual knots and Kauffman introduced the affine index polynomial of virtual knots. We introduce a zero polynomial of virtual knots of a similar type by considering weights of a suitable collection of crossings of a virtual knot diagram. We show that the zero polynomial gives a Vassiliev invariant of degree 1. It distinguishes a pair of virtual knots that cannot be distinguished by the affine index polynomial and the writhe polynomial.
In 2006 C. Hayashi gave a lower bound for the number of Reidemeister moves in deformation of two equivalent knot diagrams by using writhe and cowrithe. It can be naturally extended for two virtually isotopic virtual knot diagrams. We introduce a polynomial q K (t) of a virtual knot diagram K and give lower bounds for the number of Reidemeister moves in deformation of two virtually isotopic knots by using q K (t). We give an example which shows that the polynomial q K (t) is useful to map out a sequence of Reidemeister moves to deform a virtual knot diagram to another virtually isotopic one.
In [1], E. Appleboim introduced the notion of double dating linking class-P invariants of finite type for framed links with a fixed linking matrix P and showed that all Vassiliev link invariants are of finite type for any linking matrix and in [13], R. Trapp provided a necessary condition for a knot invariant to be a Vassiliev invariant by using twist sequences. In this paper we provide a necessary condition for a framed link invariant to be a DD-linking class-P invariant of finite type by considering a sequence of links induced from a double dating tangle. As applications we give a generalization of R. Trapp's result to see whether a link invariant is a Vassiliev invariant or not and apply the criterion for all non-zero coefficients of the Jones, HOMFLY, Q-, and Alexander polynomial.
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