L11a326

L11a325

L11a327

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Link Presentations

[edit Notes on L11a326's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X20,5,21,6 X16,9,17,10 X18,12,19,11 X22,15,9,16 X6,21,7,22 X14,8,15,7 X4,14,5,13 X8,18,1,17 X2,19,3,20
Gauss code	{1, -11, 2, -9, 3, -7, 8, -10}, {4, -1, 5, -2, 9, -8, 6, -4, 10, -5, 11, -3, 7, -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-1) (v-1) \left(v^2-v+1\right) \left(u^2 v^2-u v^2+2 u v-u+1\right)}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-4 q^{7/2}+9 q^{5/2}-15 q^{3/2}+19 \sqrt{q}-\frac{24}{\sqrt{q}}+\frac{23}{q^{3/2}}-\frac{20}{q^{5/2}}+\frac{15}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^7+4 a^3 z^5+5 a^3 z^3+2 a^3 z-a z^9-6 a z^7+z^7 a^{-1} -13 a z^5+4 z^5 a^{-1} -12 a z^3+5 z^3 a^{-1} -4 a z+2 z a^{-1} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^5-a^7 z^3+4 a^6 z^6-5 a^6 z^4+a^6 z^2+8 a^5 z^7-12 a^5 z^5+5 a^5 z^3-a^5 z+10 a^4 z^8-16 a^4 z^6+z^6 a^{-4} +10 a^4 z^4-2 z^4 a^{-4} -3 a^4 z^2+z^2 a^{-4} +8 a^3 z^9-9 a^3 z^7+4 z^7 a^{-3} +2 a^3 z^5-9 z^5 a^{-3} +a^3 z^3+5 z^3 a^{-3} -z a^{-3} +3 a^2 z^{10}+10 a^2 z^8+7 z^8 a^{-2} -31 a^2 z^6-15 z^6 a^{-2} +28 a^2 z^4+7 z^4 a^{-2} -8 a^2 z^2-z^2 a^{-2} +15 a z^9+7 z^9 a^{-1} -33 a z^7-12 z^7 a^{-1} +27 a z^5+3 z^5 a^{-1} -10 a z^3+2 a z+a z^{-1} + a^{-1} z^{-1} +3 z^{10}+7 z^8-27 z^6+22 z^4-6 z^2-1 }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     3   3 6                   6 1   -5 4                 9 3     6 2               10 6       -4 0             14 9         5 -2           11 12           1 -4         9 12             -3 -6       6 11               5 -8     3 9                 -6 -10   1 6                   5 -12   3                     -3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/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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