[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] [tricks]

$A = \begin{bmatrix}0 & 1 & 0 & \cdots & 0 \\ \vdots & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 & 1 \\ -a_{n-1} & -a_{n-2} & \cdots & \cdots & -a_{0} \end{bmatrix}$

has characteristic polynomial $z^n + a_{n-1}z^{n-1} + \dotsb + a_0$.

[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. $S = T$.

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