Knot Tricks: What Mathematical Knot Theory Can Reveal about the Structure of Khipu Knot Encoding

Abstract This article applies mathematical knot theory to the study of Andean khipus—knotted cord records, widely known for their use by the Inka empire (ca. AD 1400–1532). Despite more than 100 years of extensive study, a comprehensive understanding of the relationships and properties of different khipu knots has yet to be established. Addressing this gap, this article formalizes khipu knot relationships through the lens of mathematical knot theory, focusing on (1) common khipu knot variations that lead to misidentifications, (2) potential insights into khipu construction and data-encoding processes, and (3) the functional properties that certain knot variations offered to khipukamayuqs (khipu specialists). This article highlights the importance of recognizing topologically equivalent knots, which are visually distinct yet are structurally identical, and explores how variations of the same knot type could be used to encode or modify meaning. Notably, it reveals that several common knot forms can be transformed into visually distinct variations or other common forms without untying them, offering new perspectives on the versatility and complexity of khipus and knotted objects more broadly.