geometric topology In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originated i ...

, a branch of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the (−2, 3, 7) pretzel knot, sometimes called the Fintushel–Stern knot (after

Ron Fintushel Ronald Alan Fintushel (born 1945) is an American mathematician, specializing in low-dimensional geometric topology (specifically of 4- manifolds) and the mathematics of gauge theory. Education and career Fintushel studied mathematics at Columbia ...

and

Ronald J. Stern Ronald John Stern (born 20 January 1947) is a mathematician who works on topology, geometry, and gauge theory. He is emeritus professor at the University of California, Irvine. Stern was the first in his family to receive a college education and ...

), is an important example of a

pretzel knot A Pretzel knot may refer to: * Pretzel link: a concept in mathematics * Soft pretzel A pretzel (), from German pronunciation, standard german: Breze(l) ( and French / Alsatian: ''Bretzel'') is a type of baked bread made from dough that is c ...

which exhibits various interesting phenomena under three-dimensional and four-dimensional

surgery Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical specialty that uses operative manual and instrumental techniques on a person to investigate or treat a pat ...

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 both sailing and rock climbing as a method of stopping ropes from running out of retaining devices. Like the overhand knot, which will jam under st ...

, which has 10. All other hyperbolic knots are conjectured to have at most 6 exceptional slopes. PretzelKnot

References

External links

* {{DEFAULTSORT:-2,3,7 pretzel knot 3-manifolds 4-manifolds

Mathematical properties

References

Further reading

External links