L11a237

L11a236

L11a238

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Link Presentations

[edit Notes on L11a237's Link Presentations]

Planar diagram presentation	X8192 X12,4,13,3 X14,12,15,11 X22,15,7,16 X16,9,17,10 X10,21,11,22 X18,6,19,5 X20,18,21,17 X2738 X4,14,5,13 X6,20,1,19
Gauss code	{1, -9, 2, -10, 7, -11}, {9, -1, 5, -6, 3, -2, 10, -3, 4, -5, 8, -7, 11, -8, 6, -4}

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^2 v^4-5 u^2 v^3+8 u^2 v^2-5 u^2 v+u^2-2 u v^4+9 u v^3-15 u v^2+9 u v-2 u+v^4-5 v^3+8 v^2-5 v+1}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -11 q^{9/2}+\frac{1}{q^{9/2}}+17 q^{7/2}-\frac{3}{q^{7/2}}-23 q^{5/2}+\frac{8}{q^{5/2}}+25 q^{3/2}-\frac{15}{q^{3/2}}-q^{13/2}+5 q^{11/2}-25 \sqrt{q}+\frac{20}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +2 a z^5-3 z^5 a^{-1} +2 z^5 a^{-3} -a^3 z^3+5 a z^3-5 z^3 a^{-1} +3 z^3 a^{-3} -z^3 a^{-5} -2 a^3 z+6 a z-4 z a^{-1} +z a^{-3} -a^3 z^{-1} +3 a z^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 z^{10} a^{-2} -2 z^{10}-5 a z^9-13 z^9 a^{-1} -8 z^9 a^{-3} -5 a^2 z^8-21 z^8 a^{-2} -13 z^8 a^{-4} -13 z^8-3 a^3 z^7+2 a z^7+12 z^7 a^{-1} -4 z^7 a^{-3} -11 z^7 a^{-5} -a^4 z^6+8 a^2 z^6+46 z^6 a^{-2} +17 z^6 a^{-4} -5 z^6 a^{-6} +33 z^6+7 a^3 z^5+12 a z^5+16 z^5 a^{-1} +27 z^5 a^{-3} +15 z^5 a^{-5} -z^5 a^{-7} +3 a^4 z^4-a^2 z^4-25 z^4 a^{-2} -4 z^4 a^{-4} +4 z^4 a^{-6} -21 z^4-6 a^3 z^3-16 a z^3-21 z^3 a^{-1} -16 z^3 a^{-3} -5 z^3 a^{-5} -3 a^4 z^2-5 a^2 z^2+2 z^2 a^{-2} +3 a^3 z+10 a z+9 z a^{-1} +2 z a^{-3} +a^4+3 a^2+3-a^3 z^{-1} -3 a z^{-1} -2 a^{-1} 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     4   -4 10                   7 1   6 8                 10 4     -6 6               13 7       6 4             13 11         -2 2           12 12           0 0         9 14             5 -2       6 11               -5 -4     2 9                 7 -6   1 6                   -5 -8   2                     2 -10 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=6 }[/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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