Knot (mathematics) | Wikipedia audio article

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00:00:39 1 Formal definition
00:01:18 1.1 Projection
00:01:38 2 Types of knots
00:02:17 2.1 Tame vs. wild knots
00:02:36 2.2 Framed knot
00:03:16 2.3 Knot complement
00:03:35 2.4 JSJ decomposition
00:04:14 3 Applications to Graph Theory
00:04:54 3.1 Medial graph
00:05:33 3.2 Linkless and knotless embedding
00:06:12 4 Generalization
00:06:51 5 See also
00:07:30 6 Notes
00:07:50 7 References
00:08:29 8 External links
00:09:09 Generalization
00:09:48 S1 and M
00:10:27 0. n − j is called the codimension of the knot. An interesting aspect of Haefliger's work is that the isotopy classes of embeddings of S j in Sn form a group, with group operation given by the connect sum, provided the co-dimension is greater than two. Haefliger based his work on Stephen Smale's h-cobordism theorem. One of Smale's theorems is that when one deals with knots in co-dimension greater than two, even inequivalent knots have diffeomorphic complements. This gives the subject a different flavour than co-dimension 2 knot theory. If one allows topological or PL-isotopies, Christopher Zeeman proved that spheres do not knot when the co-dimension is greater than 2. See a generalization to manifolds.
00:11:06 See also



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SUMMARY
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In mathematics, a knot is an embedding of a circle S1 in 3-dimensional Euclidean space, R3 (also known as E3), considered up to continuous deformations (isotopies). A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term knot is also applied to embeddings of S j in Sn, especially in the case j = n − 2. The branch of mathematics that studies knots is known as knot theory, and has many simple relations to graph theory.