L10a168

L10a167

L10a169

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Link Presentations

[edit Notes on L10a168's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X18,12,19,11 X20,16,17,15 X16,20,9,19 X12,18,13,17 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -4}, {8, -5, 7, -6}, {10, -2, 5, -8, 4, -3, 6, -7}

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{(x-1) \left(u v w x-u v w-2 u v x-2 u w x+u w-u x^2+2 u x-2 v w x+v w-v x^2+2 v x+2 w x+x^2-x\right)}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{6}{q^{9/2}}-q^{7/2}+\frac{9}{q^{7/2}}+3 q^{5/2}-\frac{14}{q^{5/2}}-8 q^{3/2}+\frac{12}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}+10 \sqrt{q}-\frac{14}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^{-1} +a^5 z^{-3} -3 a^5 z-a^5 z^{-1} +3 a^3 z^3-3 a^3 z^{-3} -4 a^3 z^{-1} -z a^{-3} -a z^5+a z^3+3 a z^{-3} +2 z^3 a^{-1} - a^{-1} z^{-3} +5 a z+7 a z^{-1} -z a^{-1} -3 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^3 z^9-a z^9-3 a^4 z^8-7 a^2 z^8-4 z^8-3 a^5 z^7-10 a^3 z^7-13 a z^7-6 z^7 a^{-1} -2 a^6 z^6+3 a^2 z^6-3 z^6 a^{-2} -2 z^6-a^7 z^5+3 a^5 z^5+27 a^3 z^5+37 a z^5+13 z^5 a^{-1} -z^5 a^{-3} +3 a^6 z^4+8 a^4 z^4+21 a^2 z^4+4 z^4 a^{-2} +20 z^4+3 a^7 z^3-a^5 z^3-32 a^3 z^3-45 a z^3-15 z^3 a^{-1} +2 z^3 a^{-3} -15 a^4 z^2-36 a^2 z^2-z^2 a^{-2} -22 z^2-3 a^7 z+3 a^5 z+24 a^3 z+33 a z+14 z a^{-1} -z a^{-3} -a^6+11 a^4+24 a^2+13+a^7 z^{-1} -3 a^5 z^{-1} -12 a^3 z^{-1} -14 a z^{-1} -6 a^{-1} z^{-1} -3 a^4 z^{-2} -6 a^2 z^{-2} -3 z^{-2} +a^5 z^{-3} +3 a^3 z^{-3} +3 a z^{-3} + a^{-1} z^{-3} }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                     1 1 6                   3 1 -2 4                 5     5 2               5 3     -2 0             9 5       4 -2           7 9         2 -4         7 5           2 -6       2 7             5 -8     4 7               -3 -10   1 5                 4 -12   1                   -1 -14 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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