# New skein invariants of links

## TL;DR

This paper introduces new skein invariants for links by applying skein relations selectively to produce unlinked knots, generalizing known invariants and enabling computations solely through skein relations and initial conditions.

## Contribution

It presents a novel procedure for defining skein invariants of links that generalizes existing invariants and is computable via skein relations and initial conditions.

## Key findings

- Defines new skein invariants based on classical knot invariants
- Provides skein theoretic proofs of invariance
- Reformulates invariants as sums over sublinks

## Abstract

We introduce new skein invariants of links based on a procedure where we first apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using a 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.