Torus Knots and the Jones Polynomial

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Torus Knots and the Jones Polynomial

• Erika Ward
• Indiana University REU, Summer 2001
• Advisor Dr. Zhenghan Wang
• with Ruth Vanderpool and Danielle ODonnol

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What are Knots?

• A knot is a smooth embedding of a circle in 3
  dimensional space.
• Mental picture tangled rope with ends glued
  together
• A link is a collection of curves with the same
  properties as a knot.
• Mental picture several ropes, all tangled
  together, each with the ends attached

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What are Knots?
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When are Knots the Same?
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How you Tell Knots Apart?

• Want something thats the same for every version
  of a knot you can draw called an invariant
• No ones been entirely successful
• Most success construct a polynomial for each
  knot
• Best polynomial the Jones Polynomial

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The Jones Polynomial

• The Jones Polynomial is calculated by
  systematically taking a knot apart.
• Each crossing is broken down and factors of q are
  added following the Jones relations.
• J(unknot) 1
• J(trefoil)
• J(hopf link)

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Tying Knots to Quantum Computing

• Idea Topological Quantum Computing
• Rather than encoding data in the spin of an
  electron, encode it in the paths of particles.
• These paths can with some work be related to
  knots.
• Needed Knots with particular properties to act
  as quantum gates

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What do we Need?

• Knots that have Jones Polynomials equal to 1 at
  rth roots of unity
• An rth root of unity is a complex number that, if
  you raise to the rth power, equals 1.
• Theyre of the form .
• These knots will then be examined for particular
  properties.

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Brute Force

• Maple program that calculates the Jones
  Polynomial value at the 3rd through 10th roots of
  unity
• Generated a table for the first 251 knots, those
  with 3 through 10 crossings
• 6 interesting knots

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Torus Knots

• Can be wrapped around the surface of a torus
  think donut without any crossings
• Passes m times through the hole around the
  meridan
• Wraps n times around the hole around the
  longitude
• For a knot instead of a link m and n must be
  relatively prime

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Torus Knot (5,3)
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Jones Polynomial and Torus Knots

• Jones determined the general form of the Jones
  Polynomial for torus knots
• Given m and n, the Jones Polynomial of the torus
  knot is known.

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Theorem

• For every integer r, there are infinitely many
  torus knots with Jones Polynomial 1 at the rth
  root of unity. Further, every torus knot has
  Jones Polynomial 1 at some root of unity.

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Further

• For any r, the knot (r1, 2a(r1)1) (for any
  natural number a) has J(q) 1.
• For any knot (m,n) let r ½(m-1)(n-1). Then, at
  the rth root of unity, J(q) 1.

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What Remains to be Done?

• This gives us an infinite supply of interesting
  knotsbut doesnt tell us anything more about
  them.
• Questions
• Does this describe all of the torus knots with
  J(q) 1 at a root of unity?
• What does this tell us about non-torus knots?
• Do any of these knots provide us with quantum
  gates?

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Acknowledgements

• Research performed through Indiana University
  REU, Summer 2001
• Advisors Dr. Zhenghan Wang
• Dr. Michael Larsen
• with Ruth Vanderpool (Pacific Lutheran
  University)
• Danielle ODonnol (Indiana University)