L11a104

L11a103

L11a105

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Link Presentations

[edit Notes on L11a104's Link Presentations]

Planar diagram presentation	X6172 X2,9,3,10 X12,3,13,4 X20,11,21,12 X18,6,19,5 X22,20,5,19 X10,21,11,22 X4,17,1,18 X16,14,17,13 X14,8,15,7 X8,16,9,15
Gauss code	{1, -2, 3, -8}, {5, -1, 10, -11, 2, -7, 4, -3, 9, -10, 11, -9, 8, -5, 6, -4, 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 v^5-6 u v^4+12 u v^3-12 u v^2+4 u v+4 v^4-12 v^3+12 v^2-6 v+1}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-5 q^{7/2}+10 q^{5/2}-15 q^{3/2}+20 \sqrt{q}-\frac{23}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{19}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^7-2 a^3 z^5+4 a z^5-2 z^5 a^{-1} +a^5 z^3-6 a^3 z^3+9 a z^3-4 z^3 a^{-1} +z^3 a^{-3} +2 a^5 z-9 a^3 z+8 a z-3 z a^{-1} +2 a^5 z^{-1} -4 a^3 z^{-1} +3 a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a^2 z^{10}-2 z^{10}-5 a^3 z^9-12 a z^9-7 z^9 a^{-1} -7 a^4 z^8-12 a^2 z^8-9 z^8 a^{-2} -14 z^8-6 a^5 z^7-4 a^3 z^7+15 a z^7+8 z^7 a^{-1} -5 z^7 a^{-3} -3 a^6 z^6+8 a^4 z^6+27 a^2 z^6+21 z^6 a^{-2} -z^6 a^{-4} +38 z^6-a^7 z^5+10 a^5 z^5+24 a^3 z^5+11 a z^5+8 z^5 a^{-1} +10 z^5 a^{-3} +4 a^6 z^4-3 a^4 z^4-14 a^2 z^4-11 z^4 a^{-2} +z^4 a^{-4} -19 z^4+2 a^7 z^3-10 a^5 z^3-32 a^3 z^3-24 a z^3-7 z^3 a^{-1} -3 z^3 a^{-3} -a^6 z^2-2 a^4 z^2-4 a^2 z^2-z^2 a^{-2} -4 z^2-a^7 z+7 a^5 z+18 a^3 z+13 a z+2 z a^{-1} -z a^{-3} +2 a^4+3 a^2+ a^{-2} +3-2 a^5 z^{-1} -4 a^3 z^{-1} -3 a z^{-1} - 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     4   4 6                   6 1   -5 4                 9 4     5 2               11 6       -5 0             12 9         3 -2           11 12           1 -4         8 11             -3 -6       5 11               6 -8     3 8                 -5 -10   1 6                   5 -12   2                     -2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\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^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/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}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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