Li-Yau type inequality for curves in any codimension

For immersed curves in Euclidean space of any codimension we establish a Li--Yau type inequality that gives a lower bound of the (normalized) bending energy in terms of multiplicity. The obtained inequality is optimal for any codimension and any multiplicity except for the case of planar closed curves with odd multiplicity; in this remaining case we discover a hidden algebraic obstruction and indeed prove an exhaustive non-optimality result. The proof is mainly variational and involves Langer--Singer's classification of elasticae and Andr\'{e}'s algebraic-independence theorem for certain hypergeometric functions. We also discuss applications to elastic flows, networks, and knots.