skeinlike

Skeinlike is an adjective used primarily in knot theory and related areas of topology to describe objects, relations, or computations that resemble the skein relations that relate link diagrams which differ only in a small region. In a skein relation, the values of an invariant on diagrams L+, L−, and possibly on a smoothed diagram L0 satisfy a linear equation. A construction or invariant described as skeinlike typically satisfies similar local linear relations, though not necessarily forming a formal skein invariant on its own.

In practice, the term appears when mathematicians discuss invariants or algebraic structures that can be computed via recursions on small diagram changes, or when introducing generalized skein modules in which basis elements obey skein-like relations. The distinction between “skeinlike” and a genuine skein invariant is that the former emphasizes a resemblance to skein behavior rather than a fixed, globally defined skein relation.

The usage is common in scholarly discussions of knot invariants, tangle algebras, and lattice-model partition functions, where local changes provide a mechanism for computation. The best-known skein relations arise in the definitions of the Alexander, Jones, and Homflypt polynomials, but skeinlike wording can be used to describe new, conjectural, or context-dependent relations that mirror that pattern.

Because skeinlike is informal, its precise meaning can vary among authors and contexts. For precise terminology, one should consult the source material in which the term appears.

See also: skein relation, skein module, knot theory, link invariant, tangle algebra.