A multi-crossing (or n-crossing) is a singular point in a projection of a knot or link at which n strands cross so that each strand bisects the crossing. We generalise the classic result of Kauffman, Murasugi and Thistlethwaite relating the span of the bracket polynomial to the double-crossing number of a link, span〈K〉 ⩽ 4c
2, to the n-crossing number. We find the following lower bound on the n-crossing number in terms of the span of the bracket polynomial for any n ⩾ 3:
$$\text{span} \langle K \rangle \leq \left(\left\lfloor\frac{n^2}{2}\right\rfloor + 4n - 8\right) c_n(K).$$
We also explore
n-crossing additivity under composition, and find that for
n ⩾ 4 there are examples of knots
K
1 and
K
2 such that
cn
(
K
1#
K
2) =
cn
(
K
1) +
cn
(
K
2) − 1. Further, we present the the first extensive list of calculations of
n-crossing numbers of knots. Finally, we explore the monotonicity of the sequence of
n-crossings of a knot, which we call the crossing spectrum.