L10a88

L10a87

L10a89

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Link Presentations

[edit Notes on L10a88's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X14,6,15,5 X16,7,17,8 X18,16,19,15 X20,13,9,14 X6,17,7,18 X4,20,5,19 X2,9,3,10 X8,11,1,12
Gauss code	{1, -9, 2, -8, 3, -7, 4, -10}, {9, -1, 10, -2, 6, -3, 5, -4, 7, -5, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ -\frac{u^3 v^2-2 u^3 v+u^3+u^2 v^3-5 u^2 v^2+7 u^2 v-2 u^2-2 u v^3+7 u v^2-5 u v+u+v^3-2 v^2+v}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{5/2}-4 q^{3/2}+7 \sqrt{q}-\frac{10}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{13}{q^{5/2}}+\frac{11}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^7+2 z^3 a^5+2 z a^5+a^5 z^{-1} -z^5 a^3-z^3 a^3-2 z a^3-a^3 z^{-1} -z^5 a-z^3 a-z a+z^3 a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^5 z^9-a^3 z^9-3 a^6 z^8-7 a^4 z^8-4 a^2 z^8-3 a^7 z^7-7 a^5 z^7-11 a^3 z^7-7 a z^7-a^8 z^6+6 a^6 z^6+11 a^4 z^6-3 a^2 z^6-7 z^6+10 a^7 z^5+26 a^5 z^5+27 a^3 z^5+7 a z^5-4 z^5 a^{-1} +3 a^8 z^4+a^4 z^4+13 a^2 z^4-z^4 a^{-2} +8 z^4-10 a^7 z^3-26 a^5 z^3-20 a^3 z^3-a z^3+3 z^3 a^{-1} -2 a^8 z^2-3 a^6 z^2-5 a^4 z^2-6 a^2 z^2-2 z^2+3 a^7 z+10 a^5 z+7 a^3 z+a^4-a^5 z^{-1} -a^3 z^{-1} }[/math] (db)

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]).

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                     1 -1 4                   3   3 2                 4 1   -3 0               6 3     3 -2             7 5       -2 -4           6 5         1 -6         5 7           2 -8       4 6             -2 -10     2 6               4 -12   1 3                 -2 -14   2                   2 -16 1                     -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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