[knot theory] [invariants]

The writhe of an oriented knot is invariant under change of orientation. i.e. the writhe is an invariant of unoriented knots. (NOTE: But it’s not invariant under change of orientation for links!)

Hence the normalized Kauffman bracket polynomial of a knot is invariant under change of orientation.

[lin.alg]

Let $V$ be a $K$-vector space, then $A := \mathrm{span}\left\{ \text{linear maps}\ T : K \rightarrow V \right\} \cong V$ as vector spaces. (Abusing notation and treating $K$ as a 1D vector space over $K$ in the definition of $A$).

$A$ is a vector space. Easy to check that $\varphi : A \rightarrow V : T \mapsto Te$ where $e$ is the single basis vector in $K$ is a linear transformation. It is surjective: $\forall v \in V,\ T_v : K \rightarrow V,\ T_v(e) = v$ is a linear map. It is injective: if $Te = Se$ then $Ta = Tke = kTe = kSe = Sa\ \forall a \in K$, i.e. $S = T$.

[compl.ana]
Cross-ratio as a complete invariant on ordered 4-tuples:
Let $(z_1, z_2, z_3, z_4), (w_1, w_2, w_3, w_4)$ be 4-tuples of distinct points in $\mathbb{C}_\infty$, and define them to be equivalent if there is a Mobius transformation $f \in \mathfrak{M}$ sending one to the other.
Then $(z_1, z_2, z_3, z_4)$ and $(w_1, w_2, w_3, w_4)$ are equivalent if and only if the cross-ratios $\left[z_1, z_2; z_3, z_4\right] = \left[w_1, w_2; w_3, w_4\right]$ are the same.

Because:
$\Rightarrow$. If $(z_1, z_2, z_3, z_4), (w_1, w_2, w_3, w_4)$ are equivalent then $\left[z_1, z_2; z_3, z_4 \right] = \left[ fz_1, fz_2; fz_3, fz_4 \right] = \left[ w_1, w_2; w_3, w_4 \right]$.
$\Leftarrow$. Let $Sz = [z, z_2; z_3, z_4], Tw = [w, w_2; w_3, w_4]$. Then $T^{-1}S \in \mathfrak{M}$ sends $z_2 \rightarrow w_2, z_3 \rightarrow w_3, z_4 \rightarrow w_4$. Furthermore since $Sz_1 = Tw_1$, we have that $T^{-1}Sz_1 = w_1$.

[compl.ana]
Permuting entries of the cross-ratio: If $\lambda = \left[ z_1, z_2; z_3, z_4\right]$ then

1. $\left[ z_1, z_2; z_4, z_3\right] = \frac{1}{\lambda}$
2. $\left[ z_1, z_3; z_2, z_4\right] = 1- \lambda$
3. $\left[ z_2, z_1; z_3, z_4 \right] = \frac{1}{\lambda}$

[compl.ana]
A Mobius transformation $S = \frac{az+b}{cz+d} \neq \mathsf{id}$ has at most two fixed points in $\mathbb{C}_\infty$, with exactly one fixed point if and if only $(a-d)^2 = -4bc$.

[cat.theory]
Group multiplication is a natural transformation between the appropriate functors. Similarly the defining operations of other algebraic structures are also nat. trans. between appropriate functors.

e.g. Let $S, U : \mathbf{Grp} \rightarrow \mathbf{Set}$ where $S$ sends $G \stackrel{f}{\longrightarrow} H$ to $G^2 \stackrel{f^2}{\longrightarrow} H^2$ (the “squaring” functor), and $U$ is the forgetful functor. The set maps $\tau_G : G^2 \rightarrow G$ defined by the multiplication of $G$ are components of a natural transformation from $S$ to $U$.
(ref. Adamek-Herrlick-Strecker, Abstract and Concrete Categories, 6.2 (2).)

[alg.top]
Cell complexes are always path-connected.