topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


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topological homotopy theory

Knot theory



A homotopy between two maps f,g:XYf,g \;\colon\; X \to Y may be thought of as a path in the mapping space Map(X,Y)Map(X,Y). If, instead, one works with the space of embeddings, then one gets the concept of isotopy.

Since, for arbitrary topological spaces, the mapping space Map(X,Y)Map(X,Y) is not always well-defined as a topological space, defining a homotopy as a left homotopoy using a cylinder object, hence as a map H:X×IYH \colon X \times I \to Y is the rigorous way of saying “HH is a path from ff to gg in the mapping space.”. Similarly, the proper definition of an isotopy as a map H:X×IYH \colon X \times I \to Y with the property that H(,t):XYH(-,t) \colon X \to Y is always an embedding, is the rigorous way of saying “HH is a path from ff to gg in the embedding space.”.

In some settings, even this is not strong enough. In the notion of an ambient isotopy, one requires the path to extend to a path in the space of homeomorphisms of the ambient space.



Let XX and YY be two topological spaces. Let f,g:XYf,g \colon X \to Y be two embeddings of XX in to YY. An isotopy from ff to gg is a continuous map H:X×[0,1]YH \colon X \times [0,1] \to Y with the following properties:

  1. H(,0)=fH(-,0) = f
  2. H(,1)=gH(-,1) = g
  3. H(,t)H(-,t) is an embedding of XX in to YY for each t[0,1]t \in [0,1].

Two maps for which there exists an isotopy are said to be isotopic.


Let XX and YY be two topological spaces. Let f,g:XYf,g \colon X \to Y be two embeddings of XX in to YY. An ambient isotopy from ff to gg is a continuous map H:Y×[0,1]YH \colon Y \times [0,1] \to Y with the following properties:

  1. H(,0)H(-,0) is the identity on YY
  2. H(,1)f=gH(-,1) \circ f = g
  3. H(,t)H(-,t) is a self-homeomorphism of YY for each t[0,1]t \in [0,1].


The first two conditions on an isotopy HH imply that HH is a homotopy from ff to gg. Therefore the condition of being isotopic is stronger than that of being homotopic. As the third condition applies level-wise, the proof that homotopy is an equivalence relation carries over to show that the same is true of isotopy.


Isotopy is used where one wishes to study deformations of an object inside some ambient space that do not change the object itself. An extremely important example of this is the theory of knots and links where, to prevent unknottings and unlinkings, there have to be some restrictions on the allowed movements. These are usually encoded in terms of isotopies. It is unfortunately true, however, that a naive use of isotopy leads to strange results. If you use continuous isotopies then any two continuous embeddings of the circle into S 3S^3 are isotopic, (basically since you just pull the knot tighter and tighter, (so the knotted bit gets smaller and smaller) and at the end you just put the ‘unknot’). One way to handle this is to demand the isotopy to be piecewise linear? (or smooth), another is to work explicitly with ambient isotopy.

One of the beauties of isotopy of knots is that it can be realised very simply at the level of knot diagrams. Two knots are isotopic if their respective knot diagrams can be related using Reidemeister moves. (This is a formal theorem, but will be given elsewhere after some more development.)


General discussion includes

For isotopy in knot theory see

  • R. H Crowell and R.H. Fox, Introduction to Knot Theory, Springer, Graduate Texts 57, 1963.

  • N.D. Gilbert and T. Porter, Knots and Surfaces, Oxford U.P., 1994.

Revised on July 15, 2014 23:59:08 by Urs Schreiber (