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Next: 3.4 Integrability conditions or Up: 3. The case of Previous: 3.2 Preliminaries

Subsections

   
3.3 Constancy conditions or ${\mathcal M}_n/\delta{\mathcal M}_{n+1}$

3.3.1 Statement of the result

Definition 3.4   Let ${\mathcal Y}_n$ be the unital commutative algebra over ${\mathbb Z}$ generated by symbols Yijk for distinct indices $1\leq i,j,k\leq n$, modulo the anti-cyclicity relations Yijk=Y-1jik=Yjki.

Warning 3.5   Below we will mostly regard ${\mathcal Y}_n$ as an ${\mathbb Z}$-module, and not as an algebra. Thus we will only use the product of ${\mathcal Y}_n$ as a convenient way of writing certain elements and linear combinations of elements. The subspaces of ${\mathcal Y}_n$ that we will consider will be subspaces in the linear sense, but not ideals or subalgebras, and similarly for quotients and maps from or to ${\mathcal Y}_n$.

It is easy to define a map $\mu:{\mathcal M}_n/\delta{\mathcal M}_{n+1}\to{\mathcal Y}_n$. For an n-link L set

\begin{displaymath}\mu(L) = \prod_{1\leq i<j<k\leq n}Y_{ijk}^{\mu_{ijk}(L)}. \end{displaymath}

It follows from Section 3.2.3 that this definition descends to the quotient of ${\mathcal M}_n$ by the co-derivatives of (n+1)-links.

Theorem 4   The thus defined map $\mu:{\mathcal M}_n/\delta{\mathcal M}_{n+1}\to{\mathcal Y}_n$ is an isomorphism.

3.3.2 On a connected space, polynomials are determined by their values at any given point

3.3.3 Homotopy invariance and pure braids

3.3.4 The mask and the interchange move


 
Figure 9: A 3-mask.
\begin{figure}\begin{displaymath}
\if ny
\smash{\makebox[0pt]{\hspace{-0.5in}
...
...askDefinition.tex }
\hspace{-1.9mm}
\end{array} \end{displaymath}
\end{figure}


 
Figure 10: The co-derivative of a 3-mask.
\begin{figure}\begin{displaymath}
\if ny
\smash{\makebox[0pt]{\hspace{-0.5in}
...
...aws/deltaMask.tex }
\hspace{-1.9mm}
\end{array} \end{displaymath}
\end{figure}


 
Figure 11: The Bundle Left Twist (BLT) is the same as the Left Twist, only that the strands within each ``bundle'' are not twisted internally.
\begin{figure}\begin{displaymath}
\if ny
\smash{\makebox[0pt]{\hspace{-0.5in}
...
...BLTDefinition.tex }
\hspace{-1.9mm}
\end{array} \end{displaymath}
\end{figure}

3.3.5 Reducing third commutators

TBW.


next up previous contents
Next: 3.4 Integrability conditions or Up: 3. The case of Previous: 3.2 Preliminaries
Dror Bar-Natan
2000-03-19