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Knot Theory

Knot Theory. By Aaron Wagner Several complex variables and analytic spaces for infinite-dimensional holomorphy -Knot Theory. What is a Knot. Imagine a rope with the two ends attached together so there is no possible way for the knot to be untied.

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Knot Theory

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  1. Knot Theory By Aaron Wagner Several complex variables and analytic spaces for infinite-dimensional holomorphy -Knot Theory

  2. What is a Knot • Imagine a rope with the two ends attached together so there is no possible way for the knot to be untied. • So a knot is a one-dimensional line segment wrapping it around itself arbitrarily, and then fusing the two free ends together.

  3. Reidemeister moves • In 1926, Kurt Reidemeister proved that two knot diagrams belonging to the same knot can be related by a sequence of three Reidemeister moves.

  4. Reidemeister moves • There are three Reidemeister moves. Each one takes part of the knot and makes a change to it.

  5. Tricolorable • A knot is tricolorable if each strand of the knot diagram can be colored in one of three colors, subject to the following rules: • At least two colors must be used, and • At each crossing, the three incident strands are either all the same color or all different colors.

  6. The unknot • The Unknot is a knot that is a closed loop of string without a knot in it. • This is called the trivial knot. • It is a knot that will start out as the trivial knot, be deformed, then changed back to the trivial knot.

  7. The Unknot

  8. The Unknot • So one current problem in knot theory is to find an efficient way to figure out if any knot is equivalent to the trivial knot. • There are currently many ways to do this, but there is no way that works one hundred percent of the time.

  9. Methods So Far • There are multiple methods that can currently be used to tell if a knot is the unknot. • One way is to see if the Reidemeister moves will create the unknot.

  10. Tricolorable • If a diagram is tricolorable then it is potentially non-trivial. However there is a lot of non-trivial knots that are not 3-colorable.

  11. Other work • The Alexander polynomials distinguishes most small knots from the unknot. But this does not work for larger knots.

  12. Other work • In 1985 the Jones polynomial was created that distinguishes more knots. It is currently unknown if it always can detect the unknot. • This method produces a polynomial from any knot. This method will also always give the same polynomial for a particular knot, even if the knot looks very different. • Unfortunately it can also give identical polynomials for knots that are completely different.

  13. Other Knots • Khovanov homology was created in 1999. In 2010 Kronheimer-Mrowka stated that it will always detect the unknot, but that is still unknown to be true. • What this does is it distinguishes between any two knots that the Jones polynomial could tell apart, and some that the polynomial couldn’t. • They did this using techniques from Algebra.

  14. Other work • Combinatorial knot Floer homology was developed in 2006. It is also unknown if it always detects the unknot. • To figure this out they used symplectic geometry, a branch of geometry relating to physics. • This is used to determine whether a loop is knotted at all. It can also sometimes distinguish the unknot from any non-trivial knot.

  15. Infinitely many knots can be made, so there will always be the question of given a knot, is it the unknot?

  16. Sources • http://homepages.math.uic.edu/~kauffman/IntellUnKnot.pdf • http://www.math.ucla.edu/~cm/unknotting.pdf • http://www.cut-the-knot.org/do_you_know/knots.shtml • https://www.sciencenews.org/article/unknotting-knot-theory

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