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A Fox–Milnor theorem for knots in a thickened surface

Department of Mathematics, Trinity College, 300 Summit Street, Hartford, CT 06106, USA

https://doi.org/10.1142/S0218216519500731Cited by:0 (Source: Crossref)

Abstract

A knot in a thickened surface $K$ is a smooth embedding $K : S^{1} \to Σ \times [0, 1]$ , where $Σ$ is a closed, connected, orientable surface. There is a bijective correspondence between knots in $S^{2} \times [0, 1]$ and knots in $S^{3}$ , 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. One such famous result is the Fox–Milnor Theorem, which relates the Alexander polynomials of concordant knots. We prove a Fox–Milnor Theorem for concordant knots in a thickened surface by using Milnor torsion.

Keywords:

AMSC: 57M25, 57M27

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