## LSU Historical Dissertations and Theses

1996

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

Lawrence J. Smolinsky

#### Abstract

A polynomial invariant of links in a solid torus is defined through an algebra $H\sb{n}({1\over2}$). $H\sb{n}({1\over2}$) modulo by an ideal is the type-B Hecke algebra. This invariant satisfies the $S\sb3$-skein relation as in the 1-trivial links case of dicromatic link invariant discovered by J. Hoste and M. Kidwell. A link in the solid torus is isotopic to a closed braid which is a braid in the braid group of the annulus. We find an invariant of links through a represention $\pi$ of the braid group of the annulus to the algebra $H\sb{n}({1\over2}$). A trace map X is defined on a basis. ${\cal B} = \{(t\sp\prime\sb1)\sp{s\sb1}\cdots(t\sp\prime\sb{n})\sp{s\sb{n}} \beta\vert s\sb{i} \in \doubz,\ \beta \in H(A\sb{n-1})$, in normal form$\}$. of $H\sb{n}({1\over2}$). Then, there is a map Z from $\cup B\sb{n}(Ann)$ (braid group of annulus) to $\doubc(q,\ \sqrt{\lambda}\lbrack\tau\sb{i}\rbrack\sb{i\in\doubz}$ defined by $Z(\alpha) = (\sqrt{\lambda}z)\sp{1-x}\sqrt{\lambda}\sp{e}X(\pi(\alpha$)). The invariant $Z(\alpha$) is an ambient isotopy invariant for the links in the isotopy class that $\alpha$ represents. Therefore, this is a computational approach to the $S\sb3$-skein module for solid torus. An invariant of links in a solid torus was discovered by S. Lambropoulou through the type-B Hecke algebra. It can be recovered from $Z(\alpha$).

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