Existence of closed non-planar $p$-elasticae

We prove existence of closed non-planar $p$-elasticae for general exponents $p\in (1,\mathrm{\infty })$. In particular, we show that for any $p\in (1,\mathrm{\infty })$, there exists a countable family of non-planar $p$-elasticae, which are realized as torus knots. This generalizes well-known results of Langer–Singer for the quadratic case $p=2$ to general exponents $p\in (1,\mathrm{\infty })$.