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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

Knot Theory

By Aaron Wagner

Several complex variables and analytic spaces for infinite-dimensional holomorphy

-Knot Theory

what is a knot
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.
reidemeister moves
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.
reidemeister moves1
Reidemeister moves
  • There are three Reidemeister moves. Each one takes part of the knot and makes a change to it.
  • 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.
the unknot
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.
the unknot2
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.
methods so far
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.
  • 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.
other work
Other work
  • The Alexander polynomials distinguishes most small knots from the unknot. But this does not work for larger knots.
other work1
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.
other knots
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.
other work2
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.

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