(−2,3,7) pretzel knot
(−2,3,7) pretzel knot | |
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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 / 12n243 |
Other | |
hyperbolic, fibered, pretzel, reversible |
In geometric topology, a branch of mathematics, 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 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.
![](../I/m/PretzelKnot.jpg)
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
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