Link Invariants
Examples
Related concepts
Riemann integration, Lebesgue integration
line integral/contour integration
integration of differential forms
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
integration in ordinary differential cohomology
integration in differential K-theory
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
The Kontsevich integral generalises the Gauss integral formula? which computes the linking number of two embedded circles via integration. The Kontsevich integral is a universal Vassiliev invariant in that all Vassiliev invariants can be obtained by first applying the (final) Kontsevich integral to the knot and then applying an unframed weight system? to the result.
Let $K$ be a strict Morse knot?. Let $\widehat{\mathcal{A}}$ be the graded completion? of the algebra of chord diagrams? with $1$-term relations. The Kontsevich integral of $K$ is given by:
In this definition:
The Kontsevich integral is an invariant of Morse knots? but is not quite a knot invariant. When a “hump” is introduced to the knot then it is multiplied by $Z(H)$ where $H$ is the “humped” unknot. Therefore, it can be made in to a genuine knot invariant via the formula
where $c$ is the number of critical points of $K$. To distinguish this from the Kontsevich integral, it is sometimes called the final Kontsevich integral (and the other the preliminary one).