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(−2,3,7) pretzel knot

From Wikipedia, the free encyclopedia

(−2,3,7) pretzel knot
Pretzel knot.svg
Arf invariant 0
Crosscap no. 2
Crossing no. 12
Hyperbolic volume 2.828122088
Unknotting no. 5
Conway notation [7;-2 1;2]
Dowker notation 4, 8, -16, 2, -18, -20, -22, -24, -6, -10, -12, -14
D-T name 12n242
Last /Next 12n241<sub> </sub> / 12n243<sub> </sub>
hyperbolic, fibered, pretzel, reversible

In geometric topology, a branch of mathematics, 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 which exhibits various interesting phenomena under three-dimensional and four-dimensional surgery constructions.

Mathematical properties

The (−2, 3, 7) pretzel knot has 7 exceptional slopes, Dehn surgery slopes which give non-hyperbolic 3-manifolds. Among the enumerated knots, the only other hyperbolic knot with 7 or more is the figure-eight knot, which has 10. All other hyperbolic knots are conjectured to have at most 6 exceptional slopes.

A pretzel (−2,3,7) pretzel knot.
A pretzel (−2,3,7) pretzel knot.

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

This page was last edited on 22 January 2017, at 22:48
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