VS, TS and TG Algebras
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.
- 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 of horizontal chord chord diagrams - multiplication (vertical stacking, a binary operation) and strand permutation, strand addition, strand doubling and strand deletion (all unary operations). It is local if it satisfies the same "locality in time" and "locality in space" relations that 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 (Tangled Strands Algebra) is to tangles as a VS-algebra is to braids. Equally cryptically, it is to as a VS-algebra is to . Thus a TS-algebra has the same unary operations as a VS-algebra along with a fancier collection of "products" that allow for "reversing" and "bending back" strands before they are concatenated. Thus every TS-algebra is in particular a VS-algebra, hence if it is "local" (with the same definition as for a VS-algebra), associators and braidors make sense it it. In a TS-algebra every associator or braidor satisfying some minor further symmetry conditions leads to a knot and link invariant. Furthermore, sufficiently symmetric associators lead to full-fledged Algebraic Knot Theories.
TG-Algebras in One Paragraph
A TG-Algebra (Trivalent Graph Algebra) is to knotted trivalent graphs as a TS-algebra is to tangles. Knotted trivalent graphs are equivalent to tangles, in some topological sense; indeed, given a knotted trivalent graph, pick a maximal tree and contract it until it is just a thick point. What remains, in the complement of that think point, is just a number of knotted edges with no vertices. That is, it is a tangle. It follows that a knotted trivalent graph is merely a tangle with just a bit of extra combinatorial labeling. Likewise the notions of a TG-algebra and of a TS-algebra are nearly equivalent. They differ mostly just by how certain things are labeled.
VS-Algebras in Some Detail
The basic definitions
Definition. For a natural number let denote the set . Let denote the category whose objects are the natural numbers and whose morphisms are given by
with the obvious composition of morphisms. For brevity we will often specify morphisms/functions by simply listing their values, omitting the value at as it is anyway fixed. Thus for example, , or even shorter, , means with , , and (strictly speaking, the target space of can be any with ).
Interpretation. The object "" stands for " strands". A morphism means "for any , strand number in looks at strand number in if , and looks nowhere if ".
Definition. A VS-algebra is a contravariant functor from the category to the category of algebras over some fixed ring of scalars. We denote by and by .
Interpretation. In a TS-algebra we have an algebra for any number of strands, with multiplication corresponding to "stacking two -stranded objects (imagine braids) one on top the other". We think of an element of as " strands each of which carrying some algebraic information". If , then has strands carrying algebraic information, and if , strand number in "reads" its information from strand number in . If , strand number in reads its information from nowhere, so it carries some "default" information, presumably "empty".
You are probably familiar with the notation used elsewhere (in [Bar-Natan_97], for example) when dealing with associators and the pentagon and hexagon equations. Here's a quick dictionary:
|Elsewhere||In Words||Here||In Words|
|Copy the first strand untouched and double the second strand||The first output strand reads from the first input strand, the second and third both read from the second.|
|Delete the second strand.||Nothing reads from the second strand, as isn't in the range of .|
|Add an empty strand on the left.||The first strand reads from nowhere to it is empty. The second and third output strands read from the first and second input strands respectively.|
|Permute the strands: Install on , on and on .||Permute the strands: Read from , from and from .|
Thus the new notation is the opposite of the old when it comes to permutations: if is a permutation, is now . It is a small price to pay considering the very short description (as above) that now becomes available for VS-algebras.
The VS-Algebra of legs and VS-algebras of animals
One of the most fundamental VS-Algebras is the VS-algebra of "legs", defined as follows. Set to be the free associative and commutative algebra generated by "leg symbols" (think, " is a leg on strand number "). Then let
(thus strand number in the output looks at strand number in the input, and if it sees a leg there, it takes a copy). It is a routine exercise to verify that is indeed a VS-algebra.
An "animal" is a formal symbol with a fixed number of "legs", and it is bilinear in those legs. Thus for example we may declare that the animal has three legs, and so "a animal in " will be a symbol of the form where , and are legs in (thus ), and animals get pulled back as the sum of all ways of pulling back their legs:
|*The superscript "" stands for "non-local". A local version of this VS-algebra will be introduced after locality is introduced further below.|
Once we choose our formal symbols for animals, we may consider the free associative (though not commutative!) algebra generated by all such animals with legs in and the resulting collection of algebras and pullback operations will form a VS-algebra*.
The most standard example is the two-legged animal often referred to as "a chord from strand to strand "; this animal is also declared to be symmetric - it is declared that for all and . The resulting VS-algebra is the VS-algebra of (non-local) "horizontal chords".
The pentagon and the hexagons
|The Pentagon and the Hexagons for Parenthesized Braids|
The main thing I'd like to do in a VS-algebra is to write and solve the pentagon and hexagons equations. The unknowns in these equations are an invertible element and an invertible element , and the equations read: