A knot in a thickened surface K is a smooth embedding K : S1 --> E x [0, 1], where E is a closed, connected, orientable surface. There is a bijective correspondence between knots in S2 x [0, 1] and knots in S3, so one can view the study of knots in thickened surfaces as an extension of classical knot theory. An immediate question is if other classical definitions, concepts, and results extend or generalize to the study of knots in a thickened surface. Our motivation is to extend the well-known Fox-Milnor Theorem of classical knot theory, which relates the Alexander polynomials of concordant knots. In order to even state a similar result, we must have definitions of an Alexander polynomial and of concordance for knots in a thickened surface. Carter, Silver, and Williams have defined such an Alexander polynomial. We will describe a slightly altered definition and will compare them and discuss the advantages of our altered definition. After defining concordance of knots in a thickened surface, we state and prove a Fox-Milnor theorem for the Alexander polynomials of knots in a thickened surface. The main tools and results used in the proof come from the theory of torsion of chain complexes.
Kreinbihl, James, "A Fox-Milnor Theorem for Knots in a Thickened Surface" (2017). Dissertations. 78.
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