Concordance invariants of doubled knots and blowing up

2017/12/10 by Kim, Se-Goo, Lee, Kwan Yong
#FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1712.03486

Abstract

Let ν be either the Ozsváth-Szabó τ-invariant or the Rasmussen s-invariant, suitably normalized. For a knot K, Livingston and Naik defined the invariant t_ν(K) to be the minimum of k for which ν of the k-twisted positive Whitehead double of K vanishes. They proved that t_ν(K) is bounded above by -TB(-K), where TB is the maximal Thurston-Bennequin number. We use a blowing up process to find a crossing change formula and a new upper bound for t_ν in terms of the unknotting number. As an application, we present infinitely many knots K such that the difference between Livingston-Naik's upper bound -TB(-K) and t_ν(K) can be arbitrarily large.

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