On knot adjacency

A knot K is called n-adjacent to the unknot if it admits a projection that contains n disjoint single crossings such that changing any 0<m⩽n of these crossings, yields a projection of the unknot. Using a result of Gabai [D. Gabai, J. Differential Geom. 26 (1987) 445–503] we characterize knots that are n-adjacent to the unknot as these obtained from the unknot by n “finger moves” determined by a certain kind of trivalent graphs (Brunnian Suzuki n-graphs). Using this characterization we derive vanishing results about abelian invariants as well as Vassiliev invariants of knots that are n-adjacent to the unknot. Finally, we partially settle a conjecture of [Kalfagianni, X.-S. Lin, Preprint, 1999].