New skein invariants of links. (English) Zbl 1440.57006
Summary: We study new skein invariants of links based on a procedure where we first apply a given skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using the given invariant. A skein invariant can be computed on each link solely by the use of skein relations and a set of initial conditions. The new procedure, remarkably, leads to generalizations of the known skein invariants. We make skein invariants of classical links, \(H[R], K[Q]\) and \(D[T]\), based on the invariants of knots, \(R, Q\) and \(T\), denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. We provide skein theoretic proofs of the well-definedness of these invariants. These invariants are also reformulated into summations of the generating invariants \((R, Q, T)\) on sublinks of a given link \(L\), obtained by partitioning \(L\) into collections of sublinks. These summations exhibit the tight and surprising relationship between our generalized skein-theoretic procedure and the structure of sublinks of a given link.
MSC:
| 57K10 | Knot theory |
Keywords:
classical links; Yokonuma-Hecke algebras; mixed crossings; Reidemeister moves; stacks of knots; Homflypt polynomial; Kauffman polynomial; Dubrovnik polynomial; skein relations; skein invariants; 3-variable link invariant; closed combinatorial formulaeReferences:
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