Abstract
We present the new skein invariants of classical links, H [ H ] , K [ K ] and D [ D ] , based on the invariants of links, H, K and D, denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. The invariants are obtained by abstracting the skein relation of the corresponding invariant and making a new skein algorithm comprising two computational levels: first producing unlinked knotted components, then evaluating the resulting knots. The invariants in this paper, were revealed through the skein theoretic definition of the invariants Θ d related to the Yokonuma–Hecke algebras and their 3-variable generalization Θ , which generalizes the Homflypt polynomial. H [ H ] is the regular isotopy counterpart of Θ . The invariants K [ K ] and D [ D ] are new generalizations of the Kauffman and the Dubrovnik polynomials. We sketch skein theoretic proofs of the well-definedness and topological properties of these invariants. The invariants of this paper are reformulated into summations of the generating invariants (H, K, D) on sublinks of the given link L, obtained by partitioning L into collections of sublinks. The first such reformulation was achieved by W.B.R. Lickorish for the invariant Θ and we generalize it to the Kauffman and Dubrovnik polynomial cases. State sum models are formulated for all the invariants. These state summation models are based on our skein template algorithm which formalizes the skein theoretic process as an analogue of a statistical mechanics partition function. Relationships with statistical mechanics models are articulated. Finally, we discuss physical situations where a multi-leveled course of action is taken naturally.
Highlights
In this paper, we present the new generalized skein invariants of links, H [ H ], D [ D ] and K [K ], based on the regular isotopy version of the Homflypt polynomial, the Dubrovnik polynomial and the Kauffman polynomial, respectively (Theorems 1–3)
The generalized invariants are evaluated via a two-level procedure: for a given link we first untangle its compound knots using the skein relation of the corresponding basic invariant H, D or K and only we evaluate on unions of unlinked knots by applying a new rule, which is based on the evaluation of H, D and Symmetry 2017, 9, 226; doi:10.3390/sym9100226
We present both of these points of view and how they are related to state summations and possible relationships with statistical mechanics and applications
Summary
We present the new generalized skein invariants of links, H [ H ], D [ D ] and K [K ], based on the regular isotopy version of the Homflypt polynomial, the Dubrovnik polynomial and the Kauffman polynomial, respectively (Theorems 1–3). The combinatorial formuli (7), (15) and (16) for the generalized invariants are inspired by the Lickorish formula These closed formuli are remarkable summations of evaluations on sublinks with certain coefficients, that surprisingly satisfy the analogous mixed skein relations, so they can be regarded by themselves as definitions of the invariants H [ H ], D [ D ] and. The paper is organized as follows: In Section 2 we present the skein theoretic setting of the new skein 3-variable invariants that generalize the regular isotopy version of the Homflypt, the Dubrovnik and the Kauffman polynomials.
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