A simple proof of the strong integrality for full colored HOMFLYPT invariants

By using the HOMFLY skein theory, we prove a strong integrality theorem for the reduced colored HOMFLYPT invariants defined by a basis in the full HOMFLY skein of the annulus.

On the skein theory of dichromatic links and invariants of finite type

In [Dichromatic link invariants, Trans. Amer. Math. Soc. 321(1) (1990) 197–229], Hoste and Kidwell investigated the skein theory of oriented dichromatic links in [Formula: see text]. They introduced a multi-variable polynomial invariant [Formula: see text]. We use special substitutions for some of the parameters of the invariant [Formula: see text] to show how to deduce invariants of finite type from [Formula: see text] using partial derivatives. Then we consider the 2-component 1-trivial dichromatic links. We study the Vassiliev invariants of the 2-component in the complement of the 1-component, which is equivalent to studying Vassiliev invariants for knots in [Formula: see text] We give combinatorial formulas for the type-zero and type-one invariants and we connect these invariants to existing invariants such as Aicardi's invariant. This provides us with a topological meaning of the first partial derivative, which is also shown to be universal as a type-one invariant.

ON POWERS OF HALF-TWISTS IN M(0, 2n)

We use elementary skein theory to prove a version of a result of Stylianakis (Stylianakis, The normal closure of a power of a half-twist has infinite index in the mapping class group of a punctured sphere, arXiv:1511.02912) who showed that under mild restrictions on m and n, the normal closure of the mth power of a half-twist has infinite index in the mapping class group of a sphere with 2n punctures.

The 𝔰𝔩3 colored Jones polynomials for 2-bridge links

Kuperberg introduced web spaces for some Lie algebras which are generalizations of the Kauffman bracket skein module on a disk. We derive some formulas for [Formula: see text] and [Formula: see text] clasped web spaces by graphical calculus using skein theory. These formulas are colored version of skein relations, twist formulas and bubble skein expansion formulas. We calculate the [Formula: see text] and [Formula: see text] colored Jones polynomials of [Formula: see text]-bridge knots and links explicitly using twist formulas.

A POLYNOMIAL INVARIANT FOR LINKS IN LENS SPACES

We prove the existence of a polynomial invariant that satisfies the HOMFLY skein relation for links in a lens space. In the process we also develop a skein theory of toroidal grid diagrams in a lens space.