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

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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.

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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.

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

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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.

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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.

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The Unknot

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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.

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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.

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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.

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

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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.

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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.

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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.

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

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Sources http://homepages.math.uic.edu/~kauffman/IntellUnKnot.pdf 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 https://www.sciencenews.org/article/unknotting-knot- theory

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Section 2.3 Properties of Solution Sets

Section 2.3 Properties of Solution Sets

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