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KRISTEN MCGAHAN Knot Groups

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Group: A closed algebraic structure that has a law of composition with three properties Associative- (ab)c=a(bc) for all a,b,c in G Identity- G contains identity element I such that aI=Ia=a for all a in G Inverse- every element a in G has an inverse, another element b such that ab=I and ba=I and b is also in G

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Symmetric Group Set of all permutations Ex. The group S 3 the set of all permutations of the set (1,2,3) = ( (), (1 2)(3), (1 3)(2), (2 3)(1), (1 2 3), (1 3 2)) Each permutation can be written in cyclic notation If f(1)=3, f(2)=1, f(3)=2 this permutation is written as (1 3 2) “A set of permutations is said to generate the symmetric group if every element in the group can be written as a product of elements from the set”

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Labeling Knots Consistent- the labels of the arcs on the crossing are g, h, and k, then right hand crossings must satisfy gkg-1=h and left hand crossings ghg -1 =k Generate- labels we choose for each arc of the knot diagram must generate the group

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Labeling trefoil knot with elements from S 3 (1 2) (2 3) (1 3)

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Theorem: If a diagram for a knot can be labeled with elements from a group G, then any diagram of the knot can be so labeled with elements from that group, regardless of the choice of orientation.

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Proof of Theorem using Reidemeister Moves x y y x x yxy -1

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Proof cont. x y y x x yxy -1

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LABELING FUNDAMENTAL GROUPS WIRTINGER PRESENTATION Group Presentations of the Trefoil Knot

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Labeling Words- combination of variables and their inverses by examining each arc Equations- formed by setting the words equal to 1 give the defining relations of the group Trefoil knot

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Fundamental Group The knot group is the fundamental group of the knot complement (R 3- K) The fundamental group is the set of the product of homotopy classes (loops that have the same base point and there is a path that maps one loop to the other) Each arc in a knot diagram can be mapped to a path in the fundamental group that goes around that specific arc

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Wirtinger Presentation G(K)= (arc labels: cross relations) Start at a point near the crossing and move counterclockwise around a rectangle At each arc write down the variable if the arc is entering the crossing and the inverse of the variable if it is leaving the crossing This give us the defining relations Each relation corresponds to a loop off the knot (in the fundamental group) Trefoil knot

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Conclusions- connections between fundamental group and algebra For each arc in a knot diagram there is an element in the fundamental group This element is represented by the loop that goes around the arc Therefore, there is a map between the labelings of a knot diagram and the fundamental group of the knot complement Knot groups are useful because they are considered invariants and therefore can distinguish between knots

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