Conductive yarns in a knitted fabric may include insulating cores covered with metal layers that form signal paths. Open circuits may be formed in the yarns by removing metal from the insulating cores at selected locations within the yarns. The fabric may be formed from rows of interlocked loops of the yarn. The open circuits may be located on the loops so that each loop with an open circuit has a first segment of the metal layer that is separated from a second segment of the layer by a portion of the loop from which the metal layer has been removed. Each electrical component may have terminals that span a respective one of the open circuits and that are shorted respectively to the metal of the first and second segments.