(−2,3,7) Pretzel Knot on:
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geometric topology
In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topo ...
, a branch of mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the (−2, 3, 7) pretzel knot, sometimes called the Fintushel–Stern knot (after Ron Fintushel and Ronald J. Stern), is an important example of a pretzel knot A Pretzel knot may refer to:
* Pretzel link: a concept in mathematics
* Soft pretzel with garlic
* Stafford knot: a rope knot used in sailing and heraldry
{{Disambig ...
which exhibits various interesting phenomena under three-dimensional and four-dimensional surgery
Surgery is a medical specialty that uses manual and instrumental techniques to diagnose or treat pathological conditions (e.g., trauma, disease, injury, malignancy), to alter bodily functions (e.g., malabsorption created by bariatric surgery s ...
constructions.
Mathematical properties
The (−2, 3, 7) pretzel knot has 7 ''exceptional'' slopes,Dehn surgery
In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conceptualized as two steps: ''drilling'' then '' ...
slopes which give non-hyperbolic 3-manifold
In mathematics, more precisely in topology and differential geometry, a hyperbolic 3-manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to −1. It ...
s. Among the enumerated knots, the only other hyperbolic knot with 7 or more is the figure-eight knot
The figure-eight knot or figure-of-eight knot is a type of stopper knot. It is very important in sailing, rock climbing and caving as a method of stopping ropes from running out of retaining devices. Like the overhand knot, which will jam under ...
, which has 10. All other hyperbolic knots are conjectured to have at most 6 exceptional slopes.
Further reading
* Kirby, R., (1978). "Problems in low dimensional topology", ''Proceedings of Symposia in Pure Math.'', volume 32, 272–312. (see problem 1.77, due to Gordon, for exceptional slopes)External links
* {{DEFAULTSORT:-2, 3, 7 pretzel knot 3-manifolds