# AKT-09/HW2

Solve the following problems and submit them in class by November 3, 2009:

Problem 1. Let ${\displaystyle {\mathfrak {g}}_{1}}$ and ${\displaystyle {\mathfrak {g}}_{2}}$ be finite dimensional metrized Lie algebras, let ${\displaystyle {\mathfrak {g}}_{1}\oplus {\mathfrak {g}}_{2}}$ denote their direct sum with the obvious "orthogonal" bracket and metric, and let ${\displaystyle m}$ be the canonical isomorphism ${\displaystyle m:{\mathcal {U}}({\mathfrak {g}}_{1})\otimes {\mathcal {U}}({\mathfrak {g}}_{2})\to {\mathcal {U}}({\mathfrak {g}}_{1}\oplus {\mathfrak {g}}_{2})}$. Prove that

${\displaystyle {\mathcal {T}}_{{\mathfrak {g}}_{1}\oplus {\mathfrak {g}}_{2}}=m\circ ({\mathcal {T}}_{{\mathfrak {g}}_{1}}\otimes {\mathcal {T}}_{{\mathfrak {g}}_{2}})\circ \Box }$,

where ${\displaystyle \Box :{\mathcal {A}}(\uparrow )\to {\mathcal {A}}(\uparrow )\otimes {\mathcal {A}}(\uparrow )}$ is the co-product and ${\displaystyle {\mathcal {T}}_{\mathfrak {g}}}$ denotes the ${\displaystyle {\mathcal {U}}({\mathfrak {g}})}$-valued "tensor map" on ${\displaystyle {\mathcal {A}}}$. Can you relate this with the first problem of HW1?

Problem 2.

1. Find a concise algorithm to compute the weight system ${\displaystyle W_{so}}$ associated with the Lie algebra ${\displaystyle so(N)}$ in its defining representation.
2. Verify that your algorithm indeed satisfies the ${\displaystyle 4T}$ relation.

Problem 3. The Kauffman polynomial ${\displaystyle F(K)(a,z)}$ (see [Kauffman]) of a knot or link ${\displaystyle K}$ is ${\displaystyle a^{-w(K)}L(K)}$ where ${\displaystyle w(L)}$ is the writhe of ${\displaystyle K}$ and where ${\displaystyle L(K)}$ is the regular isotopy invariant defined by the skein relations

${\displaystyle L(s_{\pm })=a^{\pm 1}L(s))}$

(here ${\displaystyle s}$ is a strand and ${\displaystyle s_{\pm }}$ is the same strand with a ${\displaystyle \pm }$ kink added) and

$\displaystyle L(\backoverslash)+L(\slashoverback) = z\left(L(\smoothing)+L(\hsmoothing)\right)$

and by the initial condition ${\displaystyle L(\bigcirc )=1}$. State and prove the relationship between ${\displaystyle F}$ and ${\displaystyle W_{so}}$.

Mandatory but unenforced. Find yourself in the class photo and identify yourself as explained in the photo page.

[Kauffman] ^  L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 312 (1990) 417-471.