Crossing number of an alternating knot and canonical genus of its whitehead double

Abstract

A conjecture proposed by J. Tripp in 2002 and modified by T. Nakamura in 2006 states that the crossing number of any alternating knot coincides with the canonical genus of its Whitehead double. In the meantime, it has been established that this conjecture is true for a large class of alternating knots including (2,m) torus knots, 2-bridge knots, algebraic alternating knots, alternating pretzel knots and so on. In this paper, we prove that the conjecture is not true for any alternating 3-braid knot which is the connected sum of two torus knots of type (2,m) and (2,n) with odd integers m,n≥3. This results in a new modified conjecture that the crossing number of any prime alternating knot coincides with the canonical genus of its Whitehead double. We prove that this modified conjecture holds for all prime alternating 3-braid knots in addition to the known classes above.