[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.



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.

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.

\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.

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).)