Invertible knot

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In mathematics, especially in the area of topology known as knot theory, an invertible knot is a knot that can be continuously deformed to itself, but with its orientation reversed. A non-invertible knot is any knot which does not have this property. The invertibility of a knot is a knot invariant. An invertible link is the link equivalent of an invertible knot.

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[edit] Background

Number of invertible and non-invertible knots for each crossing number
Number of crossings 3 4 5 6 7 8 9 10 11 12 13 14 15 16 OEIS sequence
Non-invertible knots 0 0 0 0 0 1 2 33 187 1144 6919 38118 226581 1309875 A052402
Invertible knots 1 1 2 3 7 20 47 132 365 1032 3069 8854 26712 78830 A052403

It has long been known that most of the simple knots, such as the trefoil knot and the figure-eight knot are invertible. In 1962 Ralph Fox conjectured that some knots were non-invertible, but it was not proved that non-invertible knots exist until H. F. Trotter discovered an infinite family of pretzel knots that were non-invertible in 1963.[1] It is now known the majority of knots are non-invertible. [2]

[edit] Invertible knots

The simplest invertible knot, the trefoil knot. (Excepting the unknot)
The simplest invertible knot, the trefoil knot. (Excepting the unknot)

All knots with crossing number of 7 or less are known to be invertible. No general method is known that can distinguish if a given knot is invertible. [3]

[edit] Strongly invertible knots

A more abstract way to define an invertible knot is to say there is an orientation-preserving homeomorphism of the 3-sphere which takes the knot to itself but reverses the orientation along the knot. By imposing the stronger condition that the homeomorphism also be an involution, i.e. have period 2 in the homeomorphism group of the 3-sphere, we arrive at the definition of a strongly invertible knot. All knots with tunnel number one, such as the trefoil knot and figure-eight knot, are strongly invertible.

[edit] Non-invertible knots

The non-invertible knot 817, the simplest of the non-invertible knots.
The non-invertible knot 817, the simplest of the non-invertible knots.

The simplest example of a non-invertible knot is the knot 817 (Alexander-Briggs notation) or .2.2 (Conway notation). The pretzel knot 7, 5, 3 is non-invertible, as are all pretzel knots of the form (2p + 1), (2q + 1), (2r + 1), where p, q, and r are distinct integers, which is the infinite family proven to be non-invertible by Trotter.[4]

[edit] See also

[edit] References

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Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (May 2008)
  1. ^ Weisstein, Eric W. "Invertible Knot." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InvertibleKnot.html
  2. ^ Basic graph theory
  3. ^ Weisstein, Eric W. "Invertible Knot." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InvertibleKnot.html
  4. ^ Basic graph theory