VS, TS and TG Algebras

Contents 
Abstract
We introduce VS, TS and TGAlgebras; 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 TGAlgebras) 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 TGalgebra "plugin" and given it, will carry out all the necessary higherlevel algebra. This will make it easier to study particular cases computationally. But for this, the notions of VS, TS and TGAlgebras must first be completely specified.
VSAlgebras in One Paragraph
A VSAlgebra (Vertical Strands Algebra) is an algebraic object that is endowed with the same operations as the algebras 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 satisfies [BarNatan_97]. In any local VSalgebra 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 VSalgebra the largely undocumented braidor equations can be written and potentially be solved, and solutions always lead to braid invariants.
TSAlgebras in One Paragraph
A TSAlgebra
VSAlgebras in Detail
TSAlgebras in Detail
References
[BarNatan_97] ^ Dror BarNatan, NonAssociative Tangles, in Geometric topology, proceedings of the Georgia international topology conference, W. H. Kazez, ed., 139183, Amer. Math. Soc. and International Press, Providence, 1997.