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We derive solutions of the Kirchhoff equations for a knot tied on aninfinitely long elastic rod subjected to combined tension and twist. Weconsider the case of simple (trefoil) and double (cinquefoil) knots; other knottopologies can be investigated similarly. The rod model is based on Hookeanelasticity but is geometrically non-linear. The problem is formulated as anon-linear self-contact problem with unknown contact regions. It is solved bymeans of matched asymptotic expansions in the limit of a loose knot. Withoutany a priori assumption, we derive the topology of the contact set, whichconsists of an interval of contact flanked by two isolated points of contacts.We study the influence of the applied twist on the equilibrium.

Matched asymptotic expansions for twisted elastic knots: a self-contact problem with non- trivial contact topology