Supplementaryknot

Supplementaryknot is a concept in knot theory describing the augmentation of a base knot by a secondary component, referred to as the supplementary knot, in order to encode additional constraints or interactions within a single topological object.

A supplementaryknot consists of a base knot K embedded in three-dimensional space together with a supplementary component S, a simple closed curve disjoint from K but linked with it with a prescribed linking number L. The pair (K, S) is treated as a two-component link with designated roles: the base knot and the supplementary component. The construction emphasizes the linkage data rather than introducing intersections between the curves.

Construction can proceed by selecting K and S in general position so that their linking number is the desired L. The resulting augmented link can be depicted by a diagram, and standard link invariants extend to augmented links by incorporating the linking parameter L. Common choices include taking S to be an unknot with a specified linking number or selecting S to realize more intricate topologies.

Invariants and properties: The supplementary component influences invariants associated with the base knot. For example, augmented polynomial invariants may depend on L, providing distinctions between knots that are equivalent when considered alone. The theory also studies augmentation maps that project an augmented link back to the base knot while preserving information about the linkage.

Applications and variants: Supplementaryknots appear in modelling molecular knots with external constraints, in the study of entangled polymers, and in topological data analysis to encode paired features. They give rise to families of augmented invariants and can be simplified under certain symmetries or equivalence relations.

See also: knot theory, link, linking number, augmentation, Jones polynomial, augmented invariants.