(−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 volume3.66386[1]
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.
gollark: Ah yes, "security", because Apple's designs are perfect and those who change them will merely destroy that perfection.
gollark: I... don't really think it's bad at all, really.
gollark: 5G causes coronavirus because someone told me that on Facebook. Anything on Facebook is automatically true. QED.
gollark: When a conversation happens and you see it later, it seems to just start in some random place in the middle of it, instead of where it started or just the end of the logs.
gollark: Its scrolling does seem to be kind of weird and inconsistent.

References
  1. Agol, Ian (2010), "The minimal volume orientable hyperbolic 2-cusped 3-manifolds", Proceedings of the American Mathematical Society, 138 (10): 3723–3732, arXiv:0804.0043, doi:10.1090/S0002-9939-10-10364-5, MR 2661571.

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)
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.