2016/03/15 by JungHwan Park, Park, JungHwan
Mathematics · #57M25 #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GT #msc:57M25
paper · pdf · doi:10.48550/arxiv.1603.04740
arxiv created 2016/03/15 · openalex publication_date 2016/03/15 · arxiv updated 2016/03/16 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
Let D+(K,t) denote the positive t-twisted double of K. For a fixed integer-valued additive concordance invariant ν that bounds the smooth four genus of a knot and determines the smooth four genus of positive torus knots, Livingston and Naik defined tν(K) to be the greatest integer t such that ν(D+(K,t)) = 1. Let K1 and K2 be any knots then we prove the following inequality : tν(K1) + tν(K2) ≤ tν(K1 # K2) ≤ min(tν(K1) - tν(-K2), tν(K2) - tν(-K1)). As an application we show that tτ(K) ≠ ts(K) for infinitely many knots and that their difference can be arbitrarily large, where tτ(K) (respectively ts(K)) is tν(K) when ν is Ozváth-Szabó invariant τ (respectively when ν is normalized Rasmussen s invariant).