(−2,3,7) pretzel knot

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 invariant0
Crosscap no.2
Crossing no.12
Hyperbolic volume2.828122
Unknotting no.5
Conway notation[−2,3,7]
Dowker notation4, 8, -16, 2, -18, -20, -22, -24, -6, -10, -12, -14
D-T name12n242
Last /Next12n241  / 12n243 
Other
hyperbolic, fibered, pretzel, reversible

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.

References

    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)
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