2003/08/18 by Efstratia Kalfagianni, Xiao-Song Lin, Kalfagianni, Efstratia +1
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #math.GT
paper · pdf · doi:10.48550/arxiv.math/0308168
13 pages, 3 figures. to appear in Topology and Its Applications
arxiv created 2003/08/18 · arxiv updated 2009/12/01
A knot K is called n-adjacent to the unknot, if K admits a projection containing n generalized crossings such that changing any m (no larger than n) of them yields a projection of the unknot. We show that a non-trivial satellite knot K is n-adjacent to the unknot, for some n>0, if and only if it is n-adjacent to the unknot in any companion solid torus. In particular, every model knot of K is n-adjacent to the unknot. Along the way of proving these results, we also show that 2-bridge knots of the form Kp/q, where p/q=[2q1,2q2] for some integers q1,q2, are precisely those knots that have genus one and are 2-adjacent to the unknot.