L11a132

L11a131

L11a133

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Link Presentations

[edit Notes on L11a132's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X22,8,5,7 X20,10,21,9 X18,12,19,11 X16,14,17,13 X12,18,13,17 X10,20,11,19 X8,22,9,21 X2536 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 4, -8, 5, -7, 6, -2, 11, -6, 7, -5, 8, -4, 9, -3}

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{4 (u-1) (v-1)}{\sqrt{u} \sqrt{v}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -4 q^{9/2}+4 q^{7/2}-4 q^{5/2}+3 q^{3/2}-\frac{1}{q^{3/2}}+q^{19/2}-2 q^{17/2}+2 q^{15/2}-3 q^{13/2}+4 q^{11/2}-3 \sqrt{q}+\frac{1}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^{-1} -z^3 a^{-3} -z^3 a^{-5} -z^3 a^{-7} +a z-z a^{-1} -z a^{-7} +z a^{-9} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^8 a^{-10} -6 z^6 a^{-10} +10 z^4 a^{-10} -4 z^2 a^{-10} +2 z^9 a^{-9} -13 z^7 a^{-9} +26 z^5 a^{-9} -17 z^3 a^{-9} +2 z a^{-9} +z^{10} a^{-8} -5 z^8 a^{-8} +6 z^6 a^{-8} -z^4 a^{-8} +3 z^9 a^{-7} -17 z^7 a^{-7} +29 z^5 a^{-7} -17 z^3 a^{-7} +2 z a^{-7} +z^{10} a^{-6} -5 z^8 a^{-6} +9 z^6 a^{-6} -10 z^4 a^{-6} +4 z^2 a^{-6} +z^9 a^{-5} -3 z^7 a^{-5} +z^5 a^{-5} +z^8 a^{-4} -2 z^6 a^{-4} +z^7 a^{-3} -z^5 a^{-3} +z^6 a^{-2} +z^5 a^{-1} +a z^3+z^3 a^{-1} -2 a z-2 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +z^4-1 }[/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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 20                       1 -1 18                     1   1 16                   1 1   0 14                 2 1     1 12               2 1       -1 10             2 2         0 8           2 2           0 6         2 2             0 4       1 2               1 2     2 2                 0 0   1 3                   2 -2                         0 -4 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=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=9 }[/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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