The HOMFLY Braidor Algebra

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.

In Preparation

The information below is preliminary and cannot be trusted! (v)

This paperlet is about yet another construction of the HOMFLY polynomial, this time using "braidor equations". Though at the moment the term "braidor equations", the relationship with HOMFLY and the rationale for the whole plan is not yet described here. If you know what this is about, good. If not, bummer.

The Algebra

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^0_n=\langle S_n, x, t_1,\ldots t_n\rangle} be the free associative (but non-commutative) algebra generated by the elements of the symmetric group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_n} on $\{1,\ldots ,n\}$ and by formal variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_1\ldots t_n} , and let $A_{n}^{1}$ be the quotient of $A_{n}^{0}$ by the following "HOMFLY" relations:

1. $x$ commutes with everything else.
2. The product of permutations is as in the symmetric group $S_{n}$.
3. If $\sigma$ is a permutation then $t_{i}\sigma =\sigma t_{\sigma i}$.
4. $[t_{i},t_{j}]=x\sigma _{ij}(t_{i}-t_{j})$, where $\sigma _{ij}$ is the transposition of $i$ and $j$.

Finally, declare that $\deg x=\deg t_{i}=1$ while $\deg \sigma =0$ for every $1\leq i\leq n$ and every $\sigma \in S_{n}$, and let $A_{n}$ be the graded completion of $A_{n}^{1}$.

We say that an element of $A_{n}$ is "sorted" if it is written in the form $x^{k}\cdot \sigma t_{1}^{k_{1}}t_{2}^{k_{2}}\cdots t_{n}^{k_{n}}$ where $\sigma$ is a permutation and $k$ and the $k_{i}$'s are all non-negative integer. The HOMFLY relations imply that every element of $A_{n}$ is a linear combinations of sorted elements. Thus as a vector space, $A_{n}$ can be identified with the ring $B_{n}$ of power series in the variables $x,t_{1},\ldots ,t_{n}$ tensored with the group ring of $S_{n}$. The product of $A_{n}$ is of course very different than that of $B_{n}$.

Examples.

1. The general element of $A_{1}$ is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1)f(x,t_1)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1)} denotes the identity permutation and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x,t_1)} is a power series in two variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_1} . Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_1} is commutative.
2. The general element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_2} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (12)f(x,t_1,t_2)+(21)g(x,t_1,t_2)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} and $g$ are power series in three variables and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (12)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (21)} are the two elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_2} . Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_2} is not commutative and its product is non-trivial to describe.
3. The general element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_3} is described using Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3!=6} power series in 4 variables. The general element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_n} is described using n! power series in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} variables.

The algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_n} embeds in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{n+1}} in a trivial way by regarding Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1,\ldots,n\}} as a subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1,\ldots,n+1\}} in the obvious manner; thus when given an element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_n} we are free to think of it also as an element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{n+1}} . There is also a non-trivial map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta:A_n\to A_{n+1}} defined as follows:

1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta(x)=x} .
2. $\Delta (t_{i})=t_{i+1}+x\sigma _{1,i+1}$.
3. $\Delta$ acts on permutations by "shifting them one unit to the right", i.e., by identifying Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1,\ldots,n\}} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{2,\ldots,n+1\}\subset\{1,\ldots,n+1\}} .

The Equations

We seek to find a "braidor"; an element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_2} satisfying:

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B=(21)+x(12)+} (higher order terms).
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(\Delta B)B=(\Delta B)B(\Delta B)} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_3} .

With the vector space identification of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_n} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_n} in mind, we are seeking two power series of three variables each, whose low order behaviour is specified and which are required to satisfy 6 functional equations written in terms of 4 variables.

The Equations in Functional Form

Lemma. The following identities hold in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_n} :

1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [t_i^k, t_j] = x\sigma_{ij}(t_i^k-t_j^k)} .

A Solution

The first few terms of a solution can be computed using a computer, as shown below. But a true solution, written in a functional form, is still missing.

Computer Games

A primitive mathematica program to play with these objects is here.

A Numerology Problem

Question. Can you find nice formulas for the functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{12}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{21}} of the variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_2} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , whose Taylor expansions begin with

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{12}=x+\frac{x t_2}{3}-\frac{x t_1}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{1}{5} t_1 x^3+\frac{t_2 x^3}{5}+\frac{t_1^3 x}{45}-\frac{t_2^3 x}{45}+\frac{1}{15} t_1 t_2^2 x-\frac{1}{15} t_1^2 t_2 x}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{1}{7} t_1 x^5+\frac{t_2 x^5}{7}+\frac{11}{315} t_1^3 x^3-\frac{11}{315} t_2^3 x^3+\frac{11}{105} t_1 t_2^2 x^3-\frac{11}{105} t_1^2 t_2 x^3}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{2 t_1^5 x}{945}+\frac{2 t_2^5 x}{945}-\frac{2}{189} t_1 t_2^4 x+\frac{4}{189} t_1^2 t_2^3 x-\frac{4}{189} t_1^3 t_2^2 x+\frac{2}{189} t_1^4 t_2 x}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{1}{9} t_1 x^7+\frac{t_2 x^7}{9}+\frac{598 t_1^3 x^5}{14175}-\frac{598 t_2^3 x^5}{14175}+\frac{1619 t_1 t_2^2 x^5}{14175}-\frac{1619 t_1^2 t_2 x^5}{14175}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{74 t_1^5 x^3}{14175}+\frac{74 t_2^5 x^3}{14175}-\frac{74 t_1 t_2^4 x^3}{2835}+\frac{148 t_1^2 t_2^3 x^3}{2835}-\frac{148 t_1^3 t_2^2 x^3}{2835}+\frac{74 t_1^4 t_2 x^3}{2835}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +\frac{t_1^7 x}{4725}-\frac{t_2^7 x}{4725}+\frac{1}{675} t_1 t_2^6 x-\frac{1}{225} t_1^2 t_2^5 x+\frac{1}{135} t_1^3 t_2^4 x-\frac{1}{135} t_1^4 t_2^3 x+\frac{1}{225} t_1^5 t_2^2 x-\frac{1}{675} t_1^6 t_2 x}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{1}{11} t_1 x^9+\frac{t_2 x^9}{11}+\frac{2414 t_1^3 x^7}{51975}-\frac{2414 t_2^3 x^7}{51975}+\frac{53243 t_1 t_2^2 x^7}{467775}-\frac{53243 t_1^2 t_2 x^7}{467775}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{4058 t_1^5 x^5}{467775}+\frac{4058 t_2^5 x^5}{467775}-\frac{3904 t_1 t_2^4 x^5}{93555}+\frac{782 t_1^2 t_2^3 x^5}{10395}-\frac{782 t_1^3 t_2^2 x^5}{10395}+\frac{3904 t_1^4 t_2 x^5}{93555}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +\frac{331 t_1^7 x^3}{467775}-\frac{331 t_2^7 x^3}{467775}+\frac{331 t_1 t_2^6 x^3}{66825}-\frac{331 t_1^2 t_2^5 x^3}{22275}+\frac{331 t_1^3 t_2^4 x^3}{13365}-\frac{331 t_1^4 t_2^3 x^3}{13365}+\frac{331 t_1^5 t_2^2 x^3}{22275}-\frac{331 t_1^6 t_2 x^3}{66825}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{2 t_1^9 x}{93555}+\frac{2 t_2^9 x}{93555}-\frac{2 t_1 t_2^8 x}{10395}+\frac{8 t_1^2 t_2^7 x}{10395}-\frac{8 t_1^3 t_2^6 x}{4455}+\frac{4 t_1^4 t_2^5 x}{1485}-\frac{4 t_1^5 t_2^4 x}{1485}+\frac{8 t_1^6 t_2^3 x}{4455}-\frac{8 t_1^7 t_2^2 x}{10395}+\frac{2 t_1^8 t_2 x}{10395}}

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{21}=1+\frac{1}{9} x^2 t_1 t_2-\frac{1}{9} x^2 t_1^2 -\frac{13}{135} t_1^2 x^4+\frac{13}{135} t_1 t_2 x^4+\frac{2}{135} t_1^4 x^2+\frac{2}{45} t_1^2 t_2^2 x^2-\frac{8}{135} t_1^3 t_2 x^2}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{1147 t_1^2 x^6}{14175}+\frac{1147 t_1 t_2 x^6}{14175}+\frac{13}{525} t_1^4 x^4+\frac{878 t_1^2 t_2^2 x^4}{14175}-\frac{1229 t_1^3 t_2 x^4}{14175}-\frac{1}{525} t_1^6 x^2+\frac{2}{105} t_1^3 t_2^3 x^2-\frac{1}{35} t_1^4 t_2^2 x^2+\frac{2}{175} t_1^5 t_2 x^2}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{2939 t_1^2 x^8}{42525}+\frac{2939 t_1 t_2 x^8}{42525}+\frac{1327 t_1^4 x^6}{42525}+\frac{2896 t_1^2 t_2^2 x^6}{42525}-\frac{4223 t_1^3 t_2 x^6}{42525}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{199 t_1^6 x^4}{42525}+\frac{20}{567} t_1^3 t_2^3 x^4-\frac{97 t_1^4 t_2^2 x^4}{1701}+\frac{1124 t_1^5 t_2 x^4}{42525}+\frac{2 t_1^8 x^2}{8505}+\frac{2}{243} t_1^4 t_2^4 x^2-\frac{16 t_1^5 t_2^3 x^2}{1215}+\frac{8 t_1^6 t_2^2 x^2}{1215}-\frac{16 t_1^7 t_2 x^2}{8505}} ?