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Twist


The twist of a ribbon measures how much it twists around its axis and is defined as the integral of the incremental twist around the ribbon. A formula for the twist is given by

 Tw(K)=1/(2pi)int_Kdsepsilon_(munualpha)(dx^mu)/(ds)n^nu(dn^alpha)/(ds),
(1)

where K is parameterized by x^mu(s) for 0<=s<=L along the length of the knot by parameter s, and the frame K_f associated with K is

 y^mu=x^mu(s)+epsilonn^mu(s),
(2)

where epsilon is a small parameter and n^mu(s) is a unit vector field normal to the curve at s (Kaul 1999).

Letting Lk be the linking number of the two components of a ribbon, Tw be the twist, and Wr be the writhe, then the calugareanu theorem states that

 Lk(R)=Tw(R)+Wr(R)
(3)

(Adams 1994, p. 187).


See also

Calugareanu Theorem, Screw, Writhe

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References

Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, 1994.Kaul, R. K. "Topological Quantum Field Theories--A Meeting Ground for Physicists and Mathematicians." 15 Jul 1999. http://arxiv.org/abs/hep-th/9907119.

Referenced on Wolfram|Alpha

Twist

Cite this as:

Weisstein, Eric W. "Twist." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Twist.html

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