[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.

Scratchpad for Vector bundles & K-theory

[vect.bund]
This is a scratchpad for my summer school course in vector bundles and K-theory.

Clutching functions

Given local trivializations of a vector bundle on open sets U, V whose intersection is non-empty, these local trivializations define a clutching function \tilde{h}_{UV} : U \cap V \rightarrow GL_n(\mathbb{R}) (or GL_n(\mathbb{C})), since on the intersection the composite h_{UV} = h_V^{-1} \circ h_U : (U \cap V) \times \mathbb{R}^n \rightarrow (U \cap V) \times \mathbb{R}^n restricted to each x \in U \cap V is a linear isomorphism.

Direct sum of vector bundles

Given vector bundles p_1 : E_1 \rightarrow X, p_2 : E_2 \rightarrow X, define E_1 \oplus E_2 = \bigcup_{x\in X} (E_1)_x \oplus (E_2)_x.

  1. Fibers. The obvious thing we want for the direct sum is to let the fiber over each x be (E_1)_x \oplus (E_2)_x. So for (e_1, e_2) in the total space the projection map is (p_1 \oplus p_2)(e_1,e_2) = p_1(e_1) = p_2(e_2).
  2. We need local triviality. Define an open cover \{U_\alpha\} and local maps h_\alpha on each U_\alpha that satisfy the properties of local trivializations (except the homeomorphism bit, because we haven’t put a topology on the total space yet).
  3. Topologize. Finally, topologize the total space so that the local trivializations on U_\alpha are in fact local homeomorphisms from trivial bundles on U_\alpha to the preimage (p_1\oplus p_2)^{-1}(U_\alpha).

[alg]
Subgroups of finitely-generated abelian groups are finitely-generated.

Not necessarily true for finitely-generated non-abelian groups.