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Three-twist knot

From Wikipedia, the free encyclopedia

In knot theory, the three-twist knot is the twist knot with three-half twists. It is listed as the 52 knot[1] in the Alexander-Briggs notation, and is one of two knots with crossing number five, the other being the cinquefoil knot.

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Transcription

Properties

The three-twist knot is a prime knot, and it is invertible but not amphichiral. Its Alexander polynomial is

its Conway polynomial is

and its Jones polynomial is

[2]

Because the Alexander polynomial is not monic, the three-twist knot is not fibered.

The three-twist knot is a hyperbolic knot, with its complement having a volume of approximately 2.82812.

If the fibre of the knot in the initial image of this page were cut at the bottom right of the image, and the ends were pulled apart, it would result in a single-stranded figure-of-nine knot (not the figure-of-nine loop).

Example

Assembling of Three-twist knot.

References

  1. ^ Pinsky, Tali (1 September 2017). "On the topology of the Lorenz system". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. The Royal Society. 473 (2205): 20170374. doi:10.1098/rspa.2017.0374. PMC 5627380. PMID 28989313. Retrieved 26 August 2018. (b) the knot with three half-twists, called the 52 knot.
  2. ^ "5_2", The Knot Atlas.
This page was last edited on 9 September 2021, at 01:25
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