Fix it authentically without a knot, using this ‘slit and pull’ technique:
Cut small slits near the ends of the pieces to be joined, then thread them through each other and pull tight.
This repair is documented in medieval (6-9th cent. AD) Egyptian sandals .
Multiple ‘slit and pulls’ were also used for interweaving thongs to make decorative straps:
Anglo-Norman shoe, c. 1100AD from St. Magnus House, London (author’s reconstruction, after ).
 André J. Veldmeijer, Sandals, Shoes and Other Leatherwork from the Coptic Monastery of Deir el-Bacht: Analysis and Catalogue. Sidestone: Leiden 2011.
 Francis Grew, Margrethe de Neergaard, Susan Mitford, Shoes and Pattens: Finds from Medieval Excavations in London (Medieval Finds from Excavations in London, vol. 2). Museum of London, HMSO: London 1988.