The HOMFLY Braidor Algebra - Revision history

Drorbn: /* A Numerology Problem */

2007-02-26T19:51:18Z

A Numerology Problem

@@ Line 48: / Line 48: @@
 A primitive mathematica program to play with these objects is [http://katlas.math.toronto.edu/svn/06-1350/ here].
-==A Numerology Problem==
+==Numerology Problems==
 '''Question.''' Can you find nice formulas for the functions <math>f_{12}</math> and <math>f_{21}</math> of the variables <math>t_1</math>, <math>t_2</math> and <math>x</math>, whose Taylor expansions begin with
-<math>f_{12}=x+\frac{x t_2}{3}-\frac{x t_1}{3}</math>
+<math>f_{12}=x+\frac{x t_2}{3}-\frac{x t_1}{3}+\frac{x^3}{6}-\frac{13}{90} t_1 x^3+\frac{13 t_2 x^3}{90}+\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</math>
 :<math>-\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</math>
@@ Line 104: / Line 131: @@
 and
-<math>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}
+<math>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</math>
 :<math>-\frac{1147 t_1^2 x^6}{14175}+\frac{1147 t_1 t_2

Drorbn: /* The Equations in Functional Form */

2007-01-29T23:40:17Z

The Equations in Functional Form

← Older revision		Revision as of 19:40, 29 January 2007
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	'''Lemma.''' The following identities hold in <math>A_n</math>:		'''Lemma.''' The following identities hold in <math>A_n</math>:
	# <math>[t_i^k, t_j] = x\sigma_{ij}(t_i^k-t_j^k)</math> and therefore <math>[e^{\alpha t_i}, t_j] = x\sigma_{ij}(e^{\alpha t_i}-e^{\alpha t_j})</math>.		# <math>[t_i^k, t_j] = x\sigma_{ij}(t_i^k-t_j^k)</math> and therefore <math>[e^{\alpha t_i}, t_j] = x\sigma_{ij}(e^{\alpha t_i}-e^{\alpha t_j})</math>.
	# <math>[t_i^k, t_j^l] = x\sigma_{ij}\left(\frac{t_i^{k+l}+t_j^{k+l}-t_i^kt_j^l-t_i^lt_j^k}{t_i-t_j}\right)</math> and therefore <math>[e^{\alpha t_i}, e^{\beta t_j}] = x\sigma_{ij}\left(\frac{e^{(\alpha+\beta)t_i+e^{(\alpha+\beta)t_j}-e^{\alpha t_i+\beta t_j}-e^{\beta t_i+\alpha t_j}}{t_i-t_j}\right)</math> <br>(The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_i</math> and <math>t_j</math>, and then the "true" <math>x</math>, <math>t_i</math> and <math>t_j</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that every <math>t_i</math> occurs before any <math>t_j</math>).		# <math>[t_i^k, t_j^l] = x\sigma_{ij}\left(\frac{t_i^{k+l}+t_j^{k+l}-t_i^kt_j^l-t_i^lt_j^k}{t_i-t_j}\right)</math> and therefore <math>[e^{\alpha t_i}, e^{\beta t_j}] = x\sigma_{ij}\left(\frac{e^{(\alpha+\beta)t_i}+e^{(\alpha+\beta)t_j}-e^{\alpha t_i+\beta t_j}-e^{\beta t_i+\alpha t_j}}{t_i-t_j}\right)</math> <br>(The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_i</math> and <math>t_j</math>, and then the "true" <math>x</math>, <math>t_i</math> and <math>t_j</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that every <math>t_i</math> occurs before any <math>t_j</math>).
	# <math>\Delta(t_i^k) = t_{i+1}^k + \frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} - \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} + \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right)</math> and therefore <math>\Delta(e^{t_i}) = e^{t_{i+1}} + \frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} - \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} + \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right)</math>. (The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math>, and then the "true" <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that their subscripts form a non-decreasing sequence).		# <math>\Delta(t_i^k) = t_{i+1}^k + \frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} - \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} + \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right)</math> and therefore <math>\Delta(e^{t_i}) = e^{t_{i+1}} + \frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} - \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} + \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right)</math>. (The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math>, and then the "true" <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that their subscripts form a non-decreasing sequence).

Drorbn: /* The Equations in Functional Form */

2007-01-29T23:39:42Z

The Equations in Functional Form

← Older revision		Revision as of 19:39, 29 January 2007
Line 37:		Line 37:
	'''Lemma.''' The following identities hold in <math>A_n</math>:		'''Lemma.''' The following identities hold in <math>A_n</math>:
	# <math>[t_i^k, t_j] = x\sigma_{ij}(t_i^k-t_j^k)</math> and therefore <math>[e^{\alpha t_i}, t_j] = x\sigma_{ij}(e^{\alpha t_i}-e^{\alpha t_j})</math>.		# <math>[t_i^k, t_j] = x\sigma_{ij}(t_i^k-t_j^k)</math> and therefore <math>[e^{\alpha t_i}, t_j] = x\sigma_{ij}(e^{\alpha t_i}-e^{\alpha t_j})</math>.
	# <math>[t_i^k, t_j^l] = x\sigma_{ij}\left(\frac{t_i^{k+l}+t_j^{k+l}-t_i^kt_j^l-t_i^lt_j^k}{t_i-t_j}\right)</math> and therefore <math>[e^{\alpha t_i}, e^{\beta t_j}] = x\sigma_{ij}\left(\frac{e^{(\alpha+\beta)t_i+e^{\alpha+\beta}t_j-e^{\alpha t_i+\beta t_j}-e^{\beta t_i+\alpha t_j}}{t_i-t_j}\right)</math> <br>(The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_i</math> and <math>t_j</math>, and then the "true" <math>x</math>, <math>t_i</math> and <math>t_j</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that every <math>t_i</math> occurs before any <math>t_j</math>).		# <math>[t_i^k, t_j^l] = x\sigma_{ij}\left(\frac{t_i^{k+l}+t_j^{k+l}-t_i^kt_j^l-t_i^lt_j^k}{t_i-t_j}\right)</math> and therefore <math>[e^{\alpha t_i}, e^{\beta t_j}] = x\sigma_{ij}\left(\frac{e^{(\alpha+\beta)t_i+e^{(\alpha+\beta)t_j}-e^{\alpha t_i+\beta t_j}-e^{\beta t_i+\alpha t_j}}{t_i-t_j}\right)</math> <br>(The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_i</math> and <math>t_j</math>, and then the "true" <math>x</math>, <math>t_i</math> and <math>t_j</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that every <math>t_i</math> occurs before any <math>t_j</math>).
	# <math>\Delta(t_i^k) = t_{i+1}^k + \frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} - \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} + \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right)</math> and therefore <math>\Delta(e^{t_i}) = e^{t_{i+1}} + \frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} - \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} + \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right)</math>. (The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math>, and then the "true" <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that their subscripts form a non-decreasing sequence).		# <math>\Delta(t_i^k) = t_{i+1}^k + \frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} - \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} + \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right)</math> and therefore <math>\Delta(e^{t_i}) = e^{t_{i+1}} + \frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} - \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} + \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right)</math>. (The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math>, and then the "true" <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that their subscripts form a non-decreasing sequence).

Drorbn: /* The Equations in Functional Form */

2007-01-29T23:38:37Z

The Equations in Functional Form

← Older revision		Revision as of 19:38, 29 January 2007
Line 37:		Line 37:
	'''Lemma.''' The following identities hold in <math>A_n</math>:		'''Lemma.''' The following identities hold in <math>A_n</math>:
	# <math>[t_i^k, t_j] = x\sigma_{ij}(t_i^k-t_j^k)</math> and therefore <math>[e^{\alpha t_i}, t_j] = x\sigma_{ij}(e^{\alpha t_i}-e^{\alpha t_j})</math>.		# <math>[t_i^k, t_j] = x\sigma_{ij}(t_i^k-t_j^k)</math> and therefore <math>[e^{\alpha t_i}, t_j] = x\sigma_{ij}(e^{\alpha t_i}-e^{\alpha t_j})</math>.
	# <math>[t_i^k, t_j^l] = x\sigma_{ij}\left(\frac{t_i^{k+l}+t_j^{k+l}-t_i^kt_j^l-t_i^lt_j^k}{t_i-t_j}\right)</math> and therefore <math>[e^{\alpha t_i}, r^{\beta t_j}] = x\sigma_{ij}\left(\frac{t_i^{k+l}+t_j^{k+l}-~~t_i^kt_j~~^l-t_i~~^lt_j^k~~}{t_i-t_j}\right)</math> <br>(The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_i</math> and <math>t_j</math>, and then the "true" <math>x</math>, <math>t_i</math> and <math>t_j</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that every <math>t_i</math> occurs before any <math>t_j</math>).		# <math>[t_i^k, t_j^l] = x\sigma_{ij}\left(\frac{t_i^{k+l}+t_j^{k+l}-t_i^kt_j^l-t_i^lt_j^k}{t_i-t_j}\right)</math> and therefore <math>[e^{\alpha t_i}, e^{\beta t_j}] = x\sigma_{ij}\left(\frac{e^{(\alpha+\beta)t_i+e^{\alpha+\beta}t_j-e^{\alpha t_i+\beta t_j}-e^{\beta t_i+\alpha t_j}}{t_i-t_j}\right)</math> <br>(The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_i</math> and <math>t_j</math>, and then the "true" <math>x</math>, <math>t_i</math> and <math>t_j</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that every <math>t_i</math> occurs before any <math>t_j</math>).
	# <math>\Delta(t_i^k) = t_{i+1}^k + \frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} - \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} + \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right)</math> and therefore <math>\Delta(e^{t_i}) = e^{t_{i+1}} + \frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} - \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} + \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right)</math>. (The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math>, and then the "true" <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that their subscripts form a non-decreasing sequence).		# <math>\Delta(t_i^k) = t_{i+1}^k + \frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} - \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} + \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right)</math> and therefore <math>\Delta(e^{t_i}) = e^{t_{i+1}} + \frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} - \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} + \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right)</math>. (The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math>, and then the "true" <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that their subscripts form a non-decreasing sequence).

Drorbn: /* The Equations in Functional Form */

2007-01-29T23:36:27Z

The Equations in Functional Form

← Older revision		Revision as of 19:36, 29 January 2007
Line 37:		Line 37:
	'''Lemma.''' The following identities hold in <math>A_n</math>:		'''Lemma.''' The following identities hold in <math>A_n</math>:
	# <math>[t_i^k, t_j] = x\sigma_{ij}(t_i^k-t_j^k)</math> and therefore <math>[e^{\alpha t_i}, t_j] = x\sigma_{ij}(e^{\alpha t_i}-e^{\alpha t_j})</math>.		# <math>[t_i^k, t_j] = x\sigma_{ij}(t_i^k-t_j^k)</math> and therefore <math>[e^{\alpha t_i}, t_j] = x\sigma_{ij}(e^{\alpha t_i}-e^{\alpha t_j})</math>.
	# <math>[t_i^k, t_j^l] = x\sigma_{ij}\left(\frac{t_i^{k+l}+t_j^{k+l}-t_i^kt_j^l-t_i^lt_j^k}{t_i-t_j}\right)</math> <br>(The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_i</math> and <math>t_j</math>, and then the "true" <math>x</math>, <math>t_i</math> and <math>t_j</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that every <math>t_i</math> occurs before any <math>t_j</math>).		# <math>[t_i^k, t_j^l] = x\sigma_{ij}\left(\frac{t_i^{k+l}+t_j^{k+l}-t_i^kt_j^l-t_i^lt_j^k}{t_i-t_j}\right)</math> and therefore <math>[e^{\alpha t_i}, r^{\beta t_j}] = x\sigma_{ij}\left(\frac{t_i^{k+l}+t_j^{k+l}-t_i^kt_j^l-t_i^lt_j^k}{t_i-t_j}\right)</math> <br>(The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_i</math> and <math>t_j</math>, and then the "true" <math>x</math>, <math>t_i</math> and <math>t_j</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that every <math>t_i</math> occurs before any <math>t_j</math>).
	# <math>\Delta(t_i^k) = t_{i+1}^k + \frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} - \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} + \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right)</math> and therefore <math>\Delta(e^{t_i}) = e^{t_{i+1}} + \frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} - \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} + \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right)</math>. (The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math>, and then the "true" <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that their subscripts form a non-decreasing sequence).		# <math>\Delta(t_i^k) = t_{i+1}^k + \frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} - \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} + \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right)</math> and therefore <math>\Delta(e^{t_i}) = e^{t_{i+1}} + \frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} - \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} + \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right)</math>. (The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math>, and then the "true" <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that their subscripts form a non-decreasing sequence).

Drorbn: /* The Equations in Functional Form */

2007-01-29T23:35:36Z

The Equations in Functional Form

← Older revision		Revision as of 19:35, 29 January 2007
Line 36:		Line 36:

	'''Lemma.''' The following identities hold in <math>A_n</math>:		'''Lemma.''' The following identities hold in <math>A_n</math>:
	# <math>[t_i^k, t_j] = x\sigma_{ij}(t_i^k-t_j^k)</math> and therefore <math>[e^{t_i}, t_j] = x\sigma_{ij}(e^{t_i}-e^{t_j})</math>.		# <math>[t_i^k, t_j] = x\sigma_{ij}(t_i^k-t_j^k)</math> and therefore <math>[e^{\alpha t_i}, t_j] = x\sigma_{ij}(e^{\alpha t_i}-e^{\alpha t_j})</math>.
	# <math>[t_i^k, t_j^l] = x\sigma_{ij}\left(\frac{t_i^{k+l}+t_j^{k+l}-t_i^kt_j^l-~~t_j~~^~~kt_i~~^l}{t_i-t_j}\right)</math> (The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_i</math> and <math>t_j</math>, and then the "true" <math>x</math>, <math>t_i</math> and <math>t_j</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that every <math>t_i</math> occurs before any <math>t_j</math>).		# <math>[t_i^k, t_j^l] = x\sigma_{ij}\left(\frac{t_i^{k+l}+t_j^{k+l}-t_i^kt_j^l-t_i^lt_j^k}{t_i-t_j}\right)</math> <br>(The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_i</math> and <math>t_j</math>, and then the "true" <math>x</math>, <math>t_i</math> and <math>t_j</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that every <math>t_i</math> occurs before any <math>t_j</math>).
	# <math>\Delta(t_i^k) = t_{i+1}^k + \frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} - \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} + \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right)</math> and therefore <math>\Delta(e^{t_i}) = e^{t_{i+1}} + \frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} - \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} + \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right)</math>. (The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math>, and then the "true" <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that their subscripts form a non-decreasing sequence).		# <math>\Delta(t_i^k) = t_{i+1}^k + \frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} - \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} + \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right)</math> and therefore <math>\Delta(e^{t_i}) = e^{t_{i+1}} + \frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} - \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} + \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right)</math>. (The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math>, and then the "true" <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that their subscripts form a non-decreasing sequence).

Drorbn: /* The Equations in Functional Form */

2007-01-29T23:20:38Z

The Equations in Functional Form

← Older revision		Revision as of 19:20, 29 January 2007
Line 37:		Line 37:
	'''Lemma.''' The following identities hold in <math>A_n</math>:		'''Lemma.''' The following identities hold in <math>A_n</math>:
	# <math>[t_i^k, t_j] = x\sigma_{ij}(t_i^k-t_j^k)</math> and therefore <math>[e^{t_i}, t_j] = x\sigma_{ij}(e^{t_i}-e^{t_j})</math>.		# <math>[t_i^k, t_j] = x\sigma_{ij}(t_i^k-t_j^k)</math> and therefore <math>[e^{t_i}, t_j] = x\sigma_{ij}(e^{t_i}-e^{t_j})</math>.
			# <math>[t_i^k, t_j^l] = x\sigma_{ij}\left(\frac{t_i^{k+l}+t_j^{k+l}-t_i^kt_j^l-t_j^kt_i^l}{t_i-t_j}\right)</math> (The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_i</math> and <math>t_j</math>, and then the "true" <math>x</math>, <math>t_i</math> and <math>t_j</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that every <math>t_i</math> occurs before any <math>t_j</math>).
	# <math>\Delta(t_i^k) = t_{i+1}^k + \frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} - \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} + \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right)</math> and therefore <math>\Delta(e^{t_i}) = e^{t_{i+1}} + \frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} - \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} + \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right)</math>. (The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math>, and then the "true" <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that their subscripts form a non-decreasing sequence).		# <math>\Delta(t_i^k) = t_{i+1}^k + \frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} - \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} + \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right)</math> and therefore <math>\Delta(e^{t_i}) = e^{t_{i+1}} + \frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} - \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} + \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right)</math>. (The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math>, and then the "true" <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that their subscripts form a non-decreasing sequence).

Drorbn: /* Computer Games */

2007-01-29T00:10:59Z

Computer Games

← Older revision		Revision as of 20:10, 28 January 2007
Line 45:		Line 45:
	==Computer Games==		==Computer Games==

	A primitive mathematica program to play with these objects is [http://katlas.math.toronto.edu/svn/06-1350/~~ComputingTheJonesPolynomial.nb~~ here].		A primitive mathematica program to play with these objects is [http://katlas.math.toronto.edu/svn/06-1350/ here].

	==A Numerology Problem==		==A Numerology Problem==

Drorbn: /* A Numerology Problem */

2007-01-29T00:10:09Z

A Numerology Problem

@@ Line 80: / Line 80: @@
    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}</math>
 and
@@ Line 94: / Line 114: @@
    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}</math>?
+   x^2}{1215}-\frac{16 t_1^7 t_2 x^2}{8505}</math>

Drorbn: /* The Equations in Functional Form */

2007-01-28T02:41:57Z

The Equations in Functional Form

← Older revision		Revision as of 22:41, 27 January 2007
Line 36:		Line 36:

	'''Lemma.''' The following identities hold in <math>A_n</math>:		'''Lemma.''' The following identities hold in <math>A_n</math>:
	# <math>[t_i^k, t_j] = x\sigma_{ij}(t_i^k-t_j^k)</math> and <math>[e^{~~\alpha~~ t_i}, t_j] = x\sigma_{ij}(e^{~~\alpha~~ t_i}-e^{~~\alpha~~ t_j})</math>.		# <math>[t_i^k, t_j] = x\sigma_{ij}(t_i^k-t_j^k)</math> and therefore <math>[e^{t_i}, t_j] = x\sigma_{ij}(e^{t_i}-e^{t_j})</math>.
			# <math>\Delta(t_i^k) = t_{i+1}^k + \frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} - \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{(t_{i+1}+x)^k-t_1^k}{t_{i+1}+x-t_1} + \frac{(t_{i+1}-x)^k-t_1^k}{t_{i+1}-x-t_1}\right)</math> and therefore <math>\Delta(e^{t_i}) = e^{t_{i+1}} + \frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} - \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right) + \sigma_{1,i+1}\frac{x}{2} \left(\frac{e^{t_{i+1}+x}-e^{t_1}}{t_{i+1}+x-t_1} + \frac{e^{t_{i+1}-x}-e^{t_1}}{t_{i+1}-x-t_1}\right)</math>. (The right hand sides of these expressions should be interpreted as polynomials / power series in commuting variables <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math>, and then the "true" <math>x</math>, <math>t_1</math> and <math>t_{i+1}</math> are to be substituted in, in "normal order" - in every monomial the variables are written so that their subscripts form a non-decreasing sequence).

	==A Solution==		==A Solution==