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 Y_ijk for distinct indices $1\leq i,j,k\leq n$, modulo the anti-cyclicity relations Y_ijk=Y^-1_jik=Y_jki.

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

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

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.

 

Figure 10: The co-derivative of a 3-mask.

 

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.

3.3.5 Reducing third commutators

TBW.