The connection between homotopy groups of S2, Brunnian braids over the sphere, and Brunnian braids. This knocked me off my chair when I first heard about it. I know no conceptual explanation of this connection.
A. Berrick, F. R. Cohen, Y. L. Wong and J. Wu, Configurations, braids and homotopy groups, J. Amer. Math. Soc., 19 (2006), 265-326. Also available at http://www.math.nus.edu.sg/~matwujie/BCWWfinal.pdf See also http://www.math.nus.edu.sg/~matwujie/cohen.wu.GT.revised.29.august.2007.pdf