In geometric topology

Geometric topology

In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...

, a branch of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the (−2, 3, 7) pretzel knot, sometimes called the Fintushel–Stern knot, is an important example of a pretzel knot which exhibits various interesting phenomena under three-dimensional and four-dimensional surgery

Surgery theory

In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along...

 constructions.

Mathematical properties

The (−2, 3, 7) pretzel knot has 7 exceptional slopes, Dehn surgery

Dehn surgery

In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a specific construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link...

 slopes which give non-hyperbolic 3-manifold

Hyperbolic 3-manifold

A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously...

s. The only other hyperbolic knot with 7 or more is the figure-eight knot

Figure-eight knot (mathematics)

In knot theory, a figure-eight knot is the unique knot with a crossing number of four. This is the smallest possible crossing number except for the unknot and trefoil knot...

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