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.
| (−2,3,7) pretzel knot | |
|---|---|
 | |
| Arf invariant | 0 |
| Crosscap no. | 2 |
| Crossing no. | 12 |
| Hyperbolic volume | 2.828122 |
| Unknotting no. | 5 |
| Conway notation | [−2,3,7] |
| Dowker notation | 4, 8, -16, 2, -18, -20, -22, -24, -6, -10, -12, -14 |
| D–T notation | 12n242 |
| Last / Next | 12n241 / 12n243 |
| Other | |
| hyperbolic, fibered, pretzel, reversible | |
Mathematical properties
editThe (−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.
Further reading
edit- 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)