VS, TS and TG Algebras - Revision history

Drorbn: Reverted edit of 62.231.243.138, changed back to last version by Drorbn

2007-06-16T06:17:27Z

Reverted edit of 62.231.243.138, changed back to last version by Drorbn

@@ Line 32: / Line 32: @@
 You are probably familiar with the notation used elsewhere (in {{ref|Bar-Natan_97}}, for example) when dealing with associators and the pentagon and hexagon equations. Here's a quick dictionary:
 {| align=center cellspacing=0 border=1
+|+
 |Elsewhere
 |In Words
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 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 <math>R\in A_2</math> and an invertible element <math>\Phi\in A_3</math>, and the equations read:
-{{Equation|Pentagon|<math>(1230)^\star\Phi\cdot(1223)^\star\Phi\cdot(0123)^\star\Phi = (1123)^\star\Phi\cdot(1233)^\star\Phi</math>
+{{Equation|Pentagon|<math>(1230)^\star\Phi\cdot(1223)^\star\Phi\cdot(0123)^\star\Phi = (1123)^\star\Phi\cdot(1233)^\star\Phi</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;in&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<math>A_4</math>,}}

62.231.243.138 at 03:44, 16 June 2007

2007-06-16T03:44:50Z

@@ Line 32: / Line 32: @@
 You are probably familiar with the notation used elsewhere (in {{ref|Bar-Natan_97}}, for example) when dealing with associators and the pentagon and hexagon equations. Here's a quick dictionary:
 {| align=center cellspacing=0 border=1
-|+
 |Elsewhere
 |In Words
@@ Line 90: / Line 90: @@
 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 <math>R\in A_2</math> and an invertible element <math>\Phi\in A_3</math>, and the equations read:
-{{Equation|Pentagon|<math>(1230)^\star\Phi\cdot(1223)^\star\Phi\cdot(0123)^\star\Phi = (1123)^\star\Phi\cdot(1233)^\star\Phi</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;in&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<math>A_4</math>,}}
+{{Equation|Pentagon|<math>(1230)^\star\Phi\cdot(1223)^\star\Phi\cdot(0123)^\star\Phi = (1123)^\star\Phi\cdot(1233)^\star\Phi</math>

Drorbn: /* TS-Algebras in Some Detail */

2006-10-30T14:08:15Z

TS-Algebras in Some Detail

← Older revision		Revision as of 10:08, 30 October 2006
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	As said before, a TS-algebra is to tangles as a VS-algebra is to braids. But tangles can be composed in many more ways than braids and so in enumerating all the operations in a TS-algebra we will need a more sophisticated approach. No longer will it be enough to enumerate unary operations using a category and then to throw in a single binary operation almost as an afterthought. We now need to genuinely enumerate all <math>r</math>-ary operations for all <math>r</math>.		As said before, a TS-algebra is to tangles as a VS-algebra is to braids. But tangles can be composed in many more ways than braids and so in enumerating all the operations in a TS-algebra we will need a more sophisticated approach. No longer will it be enough to enumerate unary operations using a category and then to throw in a single binary operation almost as an afterthought. We now need to genuinely enumerate all <math>r</math>-ary operations for all <math>r</math>.

	The formal tool for enumerating <math>r</math>-ary operations is a certain cross of a category and an operad called a "[http://en.wikipedia.org/wiki/Multicategory multicategory]". In short, a multicategory has objects and identity morphisms like in an ordinary category but its sets of morphisms depend on <math>r</math> "input objects" and one "output object". Such morphisms can be composed much like ordinary <math>r</math>-ary operations, and a certain associativity of the composition maps is required.		The formal tool for enumerating <math>r</math>-ary operations is a certain cross of a category and an operad called a "[http://en.wikipedia.org/wiki/Multicategory multicategory]". In short, a multicategory has objects and identity morphisms like in an ordinary category but its sets of morphisms (which really should be called "multimorphisms") depend on <math>r</math> "input objects" and one "output object". Such morphisms can be composed much like ordinary <math>r</math>-ary operations, and a certain associativity of the composition maps is required. In a multicategory <math>{\mathbf M}</math> , we will denote the set of multimorphisms whose <math>r</math> input objects are <math>{\mathcal O}_1,\ldots,{\mathcal O}_r</math> and whose output object is <math>{\mathcal O}</math> by <math>\operatorname{mor}_{\mathbf M}({\mathcal O}_1\times\dots\times{\mathcal O_r},{\mathcal O})</math>.

			===The Multicategory '''TS'''===

			Let <math>{\mathbf{TS}}</math> be the multicategory whose objects are the natural numbers and whose morphism are

	==TG-Algebras is Some Detail==		==TG-Algebras is Some Detail==

Drorbn: /* The VS-Algebra of legs and VS-algebras of animals */

2006-10-23T16:58:52Z

The VS-Algebra of legs and VS-algebras of animals

← Older revision		Revision as of 12:58, 23 October 2006
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	Once we choose our formal symbols for animals, we may consider the free associative (though not commutative!) algebra <math>A^{nl}_n</math> generated by all such animals with legs in <math>L_n</math> and the resulting collection of algebras and pullback operations will form a VS-algebra<sup>*</sup>.		Once we choose our formal symbols for animals, we may consider the free associative (though not commutative!) algebra <math>A^{nl}_n</math> generated by all such animals with legs in <math>L_n</math> and the resulting collection of algebras and pullback operations will form a VS-algebra<sup>*</sup>.

	The most standard example is the two-legged animal <math>t_{ij}</math> often referred to as "a chord from strand <math>i</math> to strand <math>j</math>"; this animal is also declared to be symmetric - it is declared that <math>t_{ij}=t_{ji}</math> for all <math>i</math> and <math>j</math>. The resulting VS-algebra is the VS-algebra <math>A^{hor,nl}</math> of (non-local) "horizontal chords".		The most standard example is the two-legged animal <math>t_{ij}</math> often referred to as "a chord from strand <math>i</math> to strand <math>j</math>"; this animal is also declared to be symmetric - it is declared that <math>t_{ij}=t_{ji}</math> for all <math>i</math> and <math>j</math>. The resulting VS-algebra is the VS-algebra <math>{\mathcal A}^{hor,nl}</math> of (non-local) "horizontal chords".

	===The pentagon and the hexagons===		===The pentagon and the hexagons===

Drorbn: /* Locality */

2006-10-23T16:58:23Z

Locality

← Older revision		Revision as of 12:58, 23 October 2006
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	'''Definition.''' A VS-algebra <math>{\mathbf A}</math> is called "local" if <math>\Psi_1\Psi_2=\Psi_2\Psi_1</math> in <math>A_m</math> whenever "<math>\Psi_2</math> regards the strands that <math>\Psi_1</math> is supported on as equivalent". More precisely, whenever <math>\Psi_i=\alpha_i^\star\Phi_i</math> for <math>i=1,2</math>, where <math>\Phi_i\in A_{n_i}</math> and <math>\alpha_i:\underline{m}\to\underline{n_i}</math> and where <math>\alpha_2</math> is constant on the subset of <math>\underline{m}</math> on which <math>\alpha_1</math> is non-zero.		'''Definition.''' A VS-algebra <math>{\mathbf A}</math> is called "local" if <math>\Psi_1\Psi_2=\Psi_2\Psi_1</math> in <math>A_m</math> whenever "<math>\Psi_2</math> regards the strands that <math>\Psi_1</math> is supported on as equivalent". More precisely, whenever <math>\Psi_i=\alpha_i^\star\Phi_i</math> for <math>i=1,2</math>, where <math>\Phi_i\in A_{n_i}</math> and <math>\alpha_i:\underline{m}\to\underline{n_i}</math> and where <math>\alpha_2</math> is constant on the subset of <math>\underline{m}</math> on which <math>\alpha_1</math> is non-zero.

	It is an excellent idea to pause briefly and try to see why the informal part of the above definition matches with it formal part. Also, if you know about "locality in space" and "locality in scale" as they are described in {{ref\|Bar-Natan_97}}, it is an excellent idea to verify that our single notion of locality is equivalent to the conjunction of the two localities of that paper.		It is an excellent idea to pause briefly and try to see why the informal part of the above definition matches with it formal part. Also, if you know about "locality in space" and "locality in scale" as they are described in {{ref\|Bar-Natan_97}}, it is an excellent idea to verify that our single notion of locality is equivalent to the conjunction of the two localities of that paper. Finally, it is clear that every VS-algebra can be made local by imposing the relation <math>\Psi_1\Psi_2=\Psi_2\Psi_1</math> whenever necessary. Thus the VS-algebras "of animals" above also have local versions. We leave it as an exercise to our readers to verify that the local version <math>{\mathcal A}^{hor}</math> of the non-local VS-algebra <math>{\mathcal A}^{hor,nl}</math> of horizontal chords, defined above, is the well known algebra of horizontal chord diagrams modulo <math>4T</math> relations of the theory of finite type invariants of braids.

	'''Theorem.''' Any solution <math>(R,\Phi)</math> of the pentagon and hexagon equations in a local VS-algebra leads to an invariant of braids.		'''Theorem.''' Any solution <math>(R,\Phi)</math> of the pentagon and hexagon equations in a local VS-algebra leads to an invariant of braids.

Drorbn: /* The VS-Algebra of legs and VS-algebras of animals */

2006-10-23T16:51:27Z

The VS-Algebra of legs and VS-algebras of animals

← Older revision		Revision as of 12:51, 23 October 2006
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	===The VS-Algebra of legs and VS-algebras of animals===		===The VS-Algebra of legs and VS-algebras of animals===

		⚫	One of the most fundamental VS-Algebras is the VS-algebra <math>{\mathbf L}</math> of "legs", defined as follows. Set <math>L_n</math> to be the free associative and commutative algebra generated by "leg symbols" <math>\{l_1,\ldots,l_n\}</math> (think, "<math>l_k</math> is a leg on strand number <math>k</math>"). Then let
⚫	<table align=right width=25% style="border-style:solid;border-width:1px; margin: 0 0 0 1em; "><tr><td>*<small>The superscript "<math>nl</math>" stands for "non-local". A local version of this VS-algebra will be introduced after locality is introduced further below.</small></td></tr></table>
⚫	One of the most fundamental VS-Algebras is the VS-algebra <math>{\mathbf L~~}^{nl~~}</math> of "legs"~~<sup>*</sup>~~, defined as follows. Set <math>L_n</math> to be the free associative algebra generated by "leg symbols" <math>\{l_1,\ldots,l_n\}</math> (think, "<math>l_k</math> is a leg on strand number <math>k</math>"). Then let

	<center><math>(i_1i_2\cdots i_m)^\star l_k := \sum_{\{j\colon i_j=k\}}l_j</math></center>		<center><math>(i_1i_2\cdots i_m)^\star l_k := \sum_{\{j\colon i_j=k\}}l_j</math></center>

	(thus strand number <math>j</math> in the output looks at strand number <math>i_j</math> in the input, and if it sees a leg there, it takes a copy). It is a routine exercise to verify that <math>{\mathbf L~~}^{nl~~}</math> is indeed a VS-algebra.		(thus strand number <math>j</math> in the output looks at strand number <math>i_j</math> in the input, and if it sees a leg there, it takes a copy). It is a routine exercise to verify that <math>{\mathbf L}</math> 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 <math>Y</math> has three legs, and so "a <math>Y</math> animal in <math>A_n</math>" will be a symbol of the form <math>Y_{k_1k_2k_3}</math> where <math>l_{k_1}</math>, <math>l_{k_2}</math> and <math>l_{k_3}</math> are legs in <math>L_n</math> (thus <math>1\leq k_1,k_2,k_3\leq n</math>), and <math>Y</math> animals get pulled back as the sum of all ways of pulling back their legs:		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 <math>Y</math> has three legs, and so "a <math>Y</math> animal in <math>A_n</math>" will be a symbol of the form <math>Y_{k_1k_2k_3}</math> where <math>l_{k_1}</math>, <math>l_{k_2}</math> and <math>l_{k_3}</math> are legs in <math>L_n</math> (thus <math>1\leq k_1,k_2,k_3\leq n</math>), and <math>Y</math> animals get pulled back as the sum of all ways of pulling back their legs:
Line 74:		Line 73:
	<center><math>(i_1i_2\cdots i_m)^\star Y_{k_1k_2k_3} := \sum_{\{(j_1,j_2,j_3)\colon\ i_{j_1}=k_1,\ i_{j_2}=k_2, \ i_{j_3}=k_3\}}Y_{j_1j_2j_3}.</math></center>		<center><math>(i_1i_2\cdots i_m)^\star Y_{k_1k_2k_3} := \sum_{\{(j_1,j_2,j_3)\colon\ i_{j_1}=k_1,\ i_{j_2}=k_2, \ i_{j_3}=k_3\}}Y_{j_1j_2j_3}.</math></center>

		⚫	<table align=right width=25% style="border-style:solid;border-width:1px; margin: 0 0 0 1em; "><tr><td>*<small>The superscript "<math>nl</math>" stands for "non-local". A local version of this VS-algebra will be introduced after locality is introduced further below.</small></td></tr></table>
	Once we choose our formal symbols for animals, we may consider the free associative algebra <math>A^{nl}_n</math> generated by all such animals with legs in L_n and the resulting collection of algebras and pullback operations will form a VS-algebra.		Once we choose our formal symbols for animals, we may consider the free associative (though not commutative!) algebra <math>A^{nl}_n</math> generated by all such animals with legs in <math>L_n</math> and the resulting collection of algebras and pullback operations will form a VS-algebra<sup>*</sup>.

	The most standard example is the two-legged animal <math>t_{ij}</math> often referred to as "a chord from strand <math>i</math> to strand <math>j</math>"; this animal is also declared to be symmetric - it is declared that <math>t_{ij}=t_{ji}</math> for all <math>i</math> and <math>j</math>. The resulting VS-algebra is the VS-algebra <math>A^{hor,nl}</math> of (non-local) "horizontal chords".		The most standard example is the two-legged animal <math>t_{ij}</math> often referred to as "a chord from strand <math>i</math> to strand <math>j</math>"; this animal is also declared to be symmetric - it is declared that <math>t_{ij}=t_{ji}</math> for all <math>i</math> and <math>j</math>. The resulting VS-algebra is the VS-algebra <math>A^{hor,nl}</math> of (non-local) "horizontal chords".

Drorbn: /* VS-Algebras in Some Detail */

2006-10-23T16:44:17Z

VS-Algebras in Some Detail

← Older revision		Revision as of 12:44, 23 October 2006
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	Thus the new notation is the opposite of the old when it comes to permutations: if <math>\sigma</math> is a permutation, <math>\Psi^\sigma</math> is now <math>(\sigma^{-1})^\star\Psi</math>. It is a small price to pay considering the very short description (as above) that now becomes available for VS-algebras.		Thus the new notation is the opposite of the old when it comes to permutations: if <math>\sigma</math> is a permutation, <math>\Psi^\sigma</math> is now <math>(\sigma^{-1})^\star\Psi</math>. 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===

			<table align=right width=25% style="border-style:solid;border-width:1px; margin: 0 0 0 1em; "><tr><td>*<small>The superscript "<math>nl</math>" stands for "non-local". A local version of this VS-algebra will be introduced after locality is introduced further below.</small></td></tr></table>
			One of the most fundamental VS-Algebras is the VS-algebra <math>{\mathbf L}^{nl}</math> of "legs"<sup>*</sup>, defined as follows. Set <math>L_n</math> to be the free associative algebra generated by "leg symbols" <math>\{l_1,\ldots,l_n\}</math> (think, "<math>l_k</math> is a leg on strand number <math>k</math>"). Then let

			<center><math>(i_1i_2\cdots i_m)^\star l_k := \sum_{\{j\colon i_j=k\}}l_j</math></center>

			(thus strand number <math>j</math> in the output looks at strand number <math>i_j</math> in the input, and if it sees a leg there, it takes a copy). It is a routine exercise to verify that <math>{\mathbf L}^{nl}</math> 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 <math>Y</math> has three legs, and so "a <math>Y</math> animal in <math>A_n</math>" will be a symbol of the form <math>Y_{k_1k_2k_3}</math> where <math>l_{k_1}</math>, <math>l_{k_2}</math> and <math>l_{k_3}</math> are legs in <math>L_n</math> (thus <math>1\leq k_1,k_2,k_3\leq n</math>), and <math>Y</math> animals get pulled back as the sum of all ways of pulling back their legs:

			<center><math>(i_1i_2\cdots i_m)^\star Y_{k_1k_2k_3} := \sum_{\{(j_1,j_2,j_3)\colon\ i_{j_1}=k_1,\ i_{j_2}=k_2, \ i_{j_3}=k_3\}}Y_{j_1j_2j_3}.</math></center>

			Once we choose our formal symbols for animals, we may consider the free associative algebra <math>A^{nl}_n</math> generated by all such animals with legs in L_n and the resulting collection of algebras and pullback operations will form a VS-algebra.

			The most standard example is the two-legged animal <math>t_{ij}</math> often referred to as "a chord from strand <math>i</math> to strand <math>j</math>"; this animal is also declared to be symmetric - it is declared that <math>t_{ij}=t_{ji}</math> for all <math>i</math> and <math>j</math>. The resulting VS-algebra is the VS-algebra <math>A^{hor,nl}</math> of (non-local) "horizontal chords".

	===The pentagon and the hexagons===		===The pentagon and the hexagons===

Drorbn: /* TS-Algebras in Some Detail */

2006-10-18T15:16:35Z

TS-Algebras in Some Detail

← Older revision		Revision as of 11:16, 18 October 2006
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	===Enumerating ''r''-ary operations and multicategories===		===Enumerating ''r''-ary operations and multicategories===

	In the previous section we've defined VS-algebras using contravariant functors on the category <math>{\mathbf{VS}}</math>. For the purpose of our game, <math>{\mathbf{VS}}</math> is "that thing which enumerates the spaces and operations in a VS-algebras"; a VS-algebra has one space for each object of <math>{\mathbf{VS}}</math> (i.e., for each integer) and one operation for each morphism in <math>{\mathbf{VS}}</math>. More precisely, <math>{\mathbf{VS}}</math> does not enumerate ''all'' the operations in a VS-algebra - only the unary ones. As is the case for the set of braids, a VS-algebra has some, though very few, <math>r</math>-ary operations as well, corresponding to "vertical stacking". These we have included into the structure by insisting that each <math>A_n</math> space in a VS-algebra be an ~~algeba~~, hence ~~equiped~~ with a binary operation. Higher <math>r</math>-ary operations are simply iterated compositions of the binary product.		In the previous section we've defined VS-algebras using contravariant functors on the category <math>{\mathbf{VS}}</math>. For the purpose of our game, <math>{\mathbf{VS}}</math> is "that thing which enumerates the spaces and operations in a VS-algebras"; a VS-algebra has one space for each object of <math>{\mathbf{VS}}</math> (i.e., for each integer) and one operation for each morphism in <math>{\mathbf{VS}}</math>. More precisely, <math>{\mathbf{VS}}</math> does not enumerate ''all'' the operations in a VS-algebra - only the unary ones. As is the case for the set of braids, a VS-algebra has some, though very few, <math>r</math>-ary operations as well, corresponding to "vertical stacking". These we have included into the structure by insisting that each <math>A_n</math> space in a VS-algebra be an algebra, hence equipped with a binary operation. Higher <math>r</math>-ary operations are simply iterated compositions of the binary product.

	As said before, a TS-algebra is to tangles as a VS-algebra is to braids. But tangles can be composed in many more ways than braids and so in enumerating all the operations in a TS-algebra we will need a more sophisticated approach. No longer will it be enough to enumerate unary operations using a category and then to throw in a single binary operation almost as an afterthought. We now need to genuinely enumerate all <math>r</math>-ary operations for all <math>r</math>.		As said before, a TS-algebra is to tangles as a VS-algebra is to braids. But tangles can be composed in many more ways than braids and so in enumerating all the operations in a TS-algebra we will need a more sophisticated approach. No longer will it be enough to enumerate unary operations using a category and then to throw in a single binary operation almost as an afterthought. We now need to genuinely enumerate all <math>r</math>-ary operations for all <math>r</math>.

Drorbn at 15:15, 18 October 2006

2006-10-18T15:15:48Z

← Older revision		Revision as of 11:15, 18 October 2006
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	==TS-Algebras in Some Detail==		==TS-Algebras in Some Detail==

			===Enumerating ''r''-ary operations and multicategories===

			In the previous section we've defined VS-algebras using contravariant functors on the category <math>{\mathbf{VS}}</math>. For the purpose of our game, <math>{\mathbf{VS}}</math> is "that thing which enumerates the spaces and operations in a VS-algebras"; a VS-algebra has one space for each object of <math>{\mathbf{VS}}</math> (i.e., for each integer) and one operation for each morphism in <math>{\mathbf{VS}}</math>. More precisely, <math>{\mathbf{VS}}</math> does not enumerate ''all'' the operations in a VS-algebra - only the unary ones. As is the case for the set of braids, a VS-algebra has some, though very few, <math>r</math>-ary operations as well, corresponding to "vertical stacking". These we have included into the structure by insisting that each <math>A_n</math> space in a VS-algebra be an algeba, hence equiped with a binary operation. Higher <math>r</math>-ary operations are simply iterated compositions of the binary product.

			As said before, a TS-algebra is to tangles as a VS-algebra is to braids. But tangles can be composed in many more ways than braids and so in enumerating all the operations in a TS-algebra we will need a more sophisticated approach. No longer will it be enough to enumerate unary operations using a category and then to throw in a single binary operation almost as an afterthought. We now need to genuinely enumerate all <math>r</math>-ary operations for all <math>r</math>.

			The formal tool for enumerating <math>r</math>-ary operations is a certain cross of a category and an operad called a "[http://en.wikipedia.org/wiki/Multicategory multicategory]". In short, a multicategory has objects and identity morphisms like in an ordinary category but its sets of morphisms depend on <math>r</math> "input objects" and one "output object". Such morphisms can be composed much like ordinary <math>r</math>-ary operations, and a certain associativity of the composition maps is required.

	==TG-Algebras is Some Detail==		==TG-Algebras is Some Detail==

Drorbn: /* Solutions in the graded case */

2006-10-16T01:47:53Z

Solutions in the graded case

← Older revision		Revision as of 21:47, 15 October 2006
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	===Solutions in the graded case===		===Solutions in the graded case===

	If a local VS-algebra {\mathbf A} is graded in the obvious sense and if every A_n has an identity element in degree 0 one can run exactly the same iterative degree-by-degree procedure as in {{ref\|Bar-Natan_97}} in attempt to find solutions to the pentagon and hexagon. The existence and uniqueness of such solutions is then goverened by exactly the same homology theory as in {{ref\|Bar-Natan_97}}, with obvious adjustments made to accomodate for the more general setup. However, for different VS-algebras the corresponding homology groups may or may not be trivial and hence solutions may or may not exist. Thus comes our main question:		If a local VS-algebra <math>{\mathbf A}</math> is graded in the obvious sense and if every <math>A_n</math> has an identity element in degree 0 one can run exactly the same iterative degree-by-degree procedure as in {{ref\|Bar-Natan_97}} in attempt to find solutions to the pentagon and hexagon. The existence and uniqueness of such solutions is then goverened by exactly the same homology theory as in {{ref\|Bar-Natan_97}}, with obvious adjustments made to accomodate for the more general setup. However, for different VS-algebras the corresponding homology groups may or may not be trivial and hence solutions may or may not exist. Thus comes our main question:

	'''Question.''' Find examples of local VS-algebras to illuminate the various existence/uniqueness possibilities. Examples in which a pair <math>(R,\Phi)</math> exists and is unique, examples in which a solution isn't unique but the different solutions can be enumerated in a reasonable way, etc. Also, find examples of local VS-algebras in which solutions <math>(R,\Phi)</math> can be written in "closed form".		'''Question.''' Find examples of local VS-algebras to illuminate the various existence/uniqueness possibilities. Examples in which a pair <math>(R,\Phi)</math> exists and is unique, examples in which a solution isn't unique but the different solutions can be enumerated in a reasonable way, etc. Also, find examples of local VS-algebras in which solutions <math>(R,\Phi)</math> can be written in "closed form".