L11a254

L11a253

L11a255

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Link Presentations

[edit Notes on L11a254's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X4,13,5,14 X20,16,21,15 X16,8,17,7 X6,21,7,22 X18,9,19,10 X22,17,9,18 X8,20,1,19
Gauss code	{1, -2, 3, -5, 4, -8, 7, -11}, {9, -1, 2, -3, 5, -4, 6, -7, 10, -9, 11, -6, 8, -10}

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^5-2 u^3 v^4+2 u^3 v^3-u^3 v^2-u^2 v^5+3 u^2 v^4-3 u^2 v^3+3 u^2 v^2-u^2 v-u v^4+3 u v^3-3 u v^2+3 u v-u-v^3+2 v^2-2 v+1}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{10}{q^{9/2}}+\frac{8}{q^{7/2}}-\frac{7}{q^{5/2}}+\frac{4}{q^{3/2}}-\frac{1}{q^{21/2}}+\frac{3}{q^{19/2}}-\frac{5}{q^{17/2}}+\frac{7}{q^{15/2}}-\frac{9}{q^{13/2}}+\frac{10}{q^{11/2}}+\sqrt{q}-\frac{3}{\sqrt{q}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^5 z^9+a^7 z^7-7 a^5 z^7+a^3 z^7+5 a^7 z^5-17 a^5 z^5+5 a^3 z^5+7 a^7 z^3-17 a^5 z^3+6 a^3 z^3+3 a^7 z-5 a^5 z+a^5 z^{-1} -a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{13} z^3+3 a^{12} z^4-a^{12} z^2+5 a^{11} z^5-4 a^{11} z^3+a^{11} z+6 a^{10} z^6-7 a^{10} z^4+a^{10} z^2+6 a^9 z^7-9 a^9 z^5-a^9 z^3+a^9 z+6 a^8 z^8-15 a^8 z^6+8 a^8 z^4-3 a^8 z^2+5 a^7 z^9-17 a^7 z^7+17 a^7 z^5-11 a^7 z^3+4 a^7 z+2 a^6 z^{10}-2 a^6 z^8-17 a^6 z^6+27 a^6 z^4-9 a^6 z^2+8 a^5 z^9-40 a^5 z^7+61 a^5 z^5-33 a^5 z^3+5 a^5 z+a^5 z^{-1} +2 a^4 z^{10}-7 a^4 z^8-a^4 z^6+16 a^4 z^4-7 a^4 z^2-a^4+3 a^3 z^9-17 a^3 z^7+30 a^3 z^5-18 a^3 z^3+a^3 z+a^3 z^{-1} +a^2 z^8-5 a^2 z^6+7 a^2 z^4-3 a^2 z^2 }[/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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 2                       1 -1 0                     2   2 -2                   2 1   -1 -4                 5 2     3 -6               4 3       -1 -8             6 4         2 -10           4 4           0 -12         5 6             -1 -14       3 5               2 -16     2 4                 -2 -18   1 3                   2 -20   2                     -2 -22 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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