The Crossing Number of Composite Knots

One of the most basic questions in knot theory remains unresolved: is crossing number additive under connected sum? In other words, does the equality c(K1♯K2) = c(K1) + c(K2) always hold, where c(K) denotes the crossing number of a knot K and K1♯K2 is the connected sum of two (oriented) knots K1 and K2? The inequality c(K1♯K2) ≤ c(K1) + c(K2) is trivial, but very little more is known in general. Equality has been established for certain classes of knots, most notably when K1 and K2 are both alternating ([3], [6], [7]) and when K1 and K2 are both torus knots [1]. In this paper, we provide the first non-trivial lower bound on c(K1♯K2) that applies to all knots K1 and K2. Theorem 1.1. Let K1, . . . ,Kn be oriented knots in the 3-sphere. Then c(K1) + . . . + c(Kn) 152 ≤ c(K1♯ . . . ♯Kn) ≤ c(K1) + . . . + c(Kn).