VS, TS and TG Algebras

Paperlets Navigation Panel   Algebraic Knot Theory Algebraic Knot Theory - A Call for Action. The Kontsevich Integral for Knotted Trivalent Graphs. Knotted Trivalent Graphs are Finitely Presented. VS, TS and TG Algebras. The Alexander Polynomial of a Knotted Trivalent Graph. Some Questions About Trinions. Finite Type Invariants The Envelope of a Finite Type Invariant. The Envelope of The Alexander Polynomial. Some Quotients of A. Associators with Frozen Feet. The HOMFLY Braidor Algebra. Coloured Chord Diagrams. Khovanov Homology Mutation Invariance of Khovanov Homology. Khovanov Homology of Alternating Tangles. Other The Existence of the Exponential Function. The Many Faces of Alexander.

Abstract

We introduce VS-, TS- and TG-Algebras; three types of algebraic entities within which some basic equations of knot theory (related to Algebraic Knot Theory) can be written and potentially solved. Why?

• These equations are valuable yet not well understood. My hope is to study them in many simpler spaces (i.e., in many simpler VS-, TS- and TG-Algebras) than the ones that naturally occur in knot theory, in the hope that out of many test cases an understanding will emerge.
• In particular, one day I hope to write (or encourage the writing of) computer programs that will take a VS-, TS-, or TG-algebra "plug-in" and given it, will carry out all the necessary higher-level algebra. This will make it easier to study particular cases computationally. But for this, the notions of VS-, TS- and TG-Algebras must first be completely specified.

VS-Algebras in One Paragraph

A VS-Algebra (Vertical Strands Algebra) is an algebraic object that is endowed with the same operations as the algebras ${\mathcal {A}}_{n}^{hor}$ of horizontal chord chord diagrams - multiplication (vertical stacking), strand permutation, strand addition, strand doubling and strand removal. It is local if it satisfies the same "locality in time" and "locality in space" relations that ${\mathcal {A}}_{n}^{hor}$ satisfies [Bar-Natan_97]. In any local VS-algebra the equations for a Drinfel'd associator (i.e., the pentagon and the hexagon) can be written and potentially be solved, and solutions always lead to braid invariants. Likewise in any local VS-algebra the largely undocumented braidor equations can be written and potentially be solved, and solutions always lead to braid invariants.

TS-Algebras in One Paragraph

A TS-Algebra

References

[Bar-Natan_97] ^  Dror Bar-Natan, Non-Associative Tangles, in Geometric topology, proceedings of the Georgia international topology conference, W. H. Kazez, ed., 139-183, Amer. Math. Soc. and International Press, Providence, 1997.