Link Invariants
Examples
Related concepts
see also algebraic topology, functional analysis and homotopy theory
topological space (see also locale)
fiber space, space attachment
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
subsets are closed in a closed subspace precisely if they are closed in the ambient space
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
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
Theorems
The Reidemeister moves are elementary, local changes to knot or link diagrams preserving the topology.
These moves were given in the book by Reidemeister.
We define the Reidemeister moves in terms of link diagrams.
The conventions are that part of the diagram is shown with the ‘move’ indicated, but that outside that, the diagram of the link is unchanged and that no other part of the diagram appears at the crossings that are changed by that move.
In manipulating a link diagram, the changes obtained by elongating part of the diagram, rotating it, deforming the plane, etc., are not counted as changing the structure of the diagram provided they do not change any crossing, and so they may be done without comment. The ‘moves’ outlined here do change things locally near crossings.
The local configurations
are interchangable;
The configurations
are interchangeable;
The configurations
are interchangeable.
Often the terminology is used that two link diagrams, $D$ and $D'$ are isotopic or isotopic by moves if $D$ can be transformed into $D'$ by some sequence of the three Reidemeister moves, R1, R2 and R3. They are called regularly isotopic if the transformation can be done without using R1. It is clear that both isotopy and regular isotopy give equivalence relations on link diagrams.
The following theorem is a form of Reidemeister’s theorem:
Two knot diagrams correspond to isotopic knots precisely if they can be connected by a sequence of applications of the three Reidemeister moves.
The corresponding statement for links and link diagram is also true.
Because of this, we can hope to use the Reidemeister moves to verify invariance of a potential link invariant. For instance, as none of the moves alters the number of components of a link diagram, the number of components is an isotopy invariant (which is not at all surprising!) We can conclude that the Hopf link and the Borromean rings are not isotopic.
A deeper observation is that the linking number is a link invariant, for which see that entry.
3-colourability is a knot invariant as is easy to check. This shows, for instance, that the trefoil knot is not equivalent to the unknot, and hence shows, very simply, that non-trivial knots exist. It also shows that the figure eight knot is not equivalent to the trefoil.
(Links to other instances of this sort of argument will get put here later on.)
One of the original sources is:
Simple examples of invariants under the Reidemeister Moves are given in, for instance,
Most modern texts on Knot Theory will contain discussions of the Reidemeister Moves.