2019/05/09 by James Kreinbihl, Kreinbihl, James · 1 citation
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #math.GT
paper · pdf · doi:10.48550/arxiv.1905.03595
arxiv created 2019/05/09 · arxiv updated 2019/05/10
A knot in a thickened surface K is a smooth embedding K:S1 → Σ× [0,1], where Σ is a closed, connected, orientable surface. There is a bijective correspondence between knots in S2 × [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. 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.