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Knot Groups and Their Homomorphisms into SU(2)
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We begin with an introduction to algebraic topology, knot theory, and SU(2) matrices as a subset of the quaternions, then proceed to introduce a technique of finding homomorphisms of knot complement fundamental groups into SU(2) and illustrate it by finding homomorphisms for the fundamental groups of the complement of the trefoil and the complement of the Whitehead link. Finally, the Seifert-van Kampen theorem allows us to find pairs of those homomorphisms, with nonabelian image, which give rise to homomorphisms from the knot group of the Whitehead double and therefore prove that the Whitehead double of the trefoil knot is not the trivial knot.