Doctor of Philosophy (PhD)
By a knot, or link, we mean a circle, or a collection of circles, embedded in the three-sphere S3. The study of knots is a very rich subject and plays a key role in the area of low-dimensional topology. In fact, a theorem of W.B.R. Lickorish and A.D. Wallace states that any three-dimensional manifold may be described by Dehn surgery along a link which is the process of removing the link from S3 and then gluing it back in a way that possibly changes the resulting manifold. In this dissertation, we will be interested in the pair (K, ρ) consisting of a knot K and a surjective map ρ from the knot group onto a dihedral group of order 2p called a coloring. Such an object is said to be a p-colored knot. In "Surgery untying of colored knots", D. Moskovich conjectures that for any odd prime p there are exactly p equivalence classes of p-colored knots up to surgery which preserves colorability. This is an analog to the classical result that every knot has a "surgery description" or equivalently that every knot is surgery equivalent to the unknot if we place fewer restrictions on the allowed surgery curves. We show that there are at most 2p equivalence classes for p any odd number. This is an improvement upon the previous results by Moskovich for p = 3, and 5, with no upper bound given in general. We do this by defining a new invariant, or an algebraic object associated to a p-colored knot, which is "complete" in the sense that two p-colored knots are surgery equivalent if and only if they both have the same value of this invariant. The complete invariant consists of Moskovich’s "colored untying invariant" redefined in the same way as the three-manifold invariants developed by T. Cochran, A. Gerges, and K. Orr, and another object we call the η invariant. We also extend these methods to give similar results for "A4-colored knots" which have representations onto the alternating group on four letters.
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Wallace, Steven Daniel, "Surgery description of colored knots" (2008). LSU Doctoral Dissertations. 2324.
Litherland, Richard A.