L11a131

L11a130

L11a132

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Link Presentations

[edit Notes on L11a131's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X20,8,21,7 X22,10,5,9 X8,22,9,21 X18,12,19,11 X16,14,17,13 X12,18,13,17 X10,20,11,19 X2536 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -5, 4, -9, 6, -8, 7, -2, 11, -7, 8, -6, 9, -3, 5, -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-1) (v-1) \left(3 v^2-2 v+3\right)}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{21/2}-3 q^{19/2}+5 q^{17/2}-7 q^{15/2}+9 q^{13/2}-10 q^{11/2}+10 q^{9/2}-8 q^{7/2}+5 q^{5/2}-4 q^{3/2}+\sqrt{q}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{-9} +z a^{-9} -z^5 a^{-7} -2 z^3 a^{-7} -z a^{-7} -z^5 a^{-5} -z^3 a^{-5} +z a^{-5} + a^{-5} z^{-1} -z^5 a^{-3} -3 z^3 a^{-3} -4 z a^{-3} -3 a^{-3} z^{-1} +z^3 a^{-1} +3 z a^{-1} +2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^{10} a^{-6} -z^{10} a^{-8} -z^9 a^{-5} -4 z^9 a^{-7} -3 z^9 a^{-9} -z^8 a^{-4} +3 z^8 a^{-6} -4 z^8 a^{-10} -z^7 a^{-3} +z^7 a^{-5} +15 z^7 a^{-7} +10 z^7 a^{-9} -3 z^7 a^{-11} -z^6 a^{-2} -8 z^6 a^{-6} +6 z^6 a^{-8} +14 z^6 a^{-10} -z^6 a^{-12} -z^5 a^{-1} -z^5 a^{-3} -3 z^5 a^{-5} -24 z^5 a^{-7} -11 z^5 a^{-9} +10 z^5 a^{-11} +z^4 a^{-2} -z^4 a^{-4} +6 z^4 a^{-6} -7 z^4 a^{-8} -12 z^4 a^{-10} +3 z^4 a^{-12} +4 z^3 a^{-1} +7 z^3 a^{-3} +3 z^3 a^{-5} +13 z^3 a^{-7} +7 z^3 a^{-9} -6 z^3 a^{-11} +3 z^2 a^{-2} +5 z^2 a^{-4} -z^2 a^{-6} +z^2 a^{-8} +3 z^2 a^{-10} -z^2 a^{-12} -5 z a^{-1} -7 z a^{-3} -2 z a^{-5} -2 z a^{-7} -2 z a^{-9} -3 a^{-2} -3 a^{-4} - a^{-6} +2 a^{-1} z^{-1} +3 a^{-3} z^{-1} + a^{-5} 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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 22                       1 -1 20                     2   2 18                   3 1   -2 16                 4 2     2 14               5 3       -2 12             5 4         1 10           5 5           0 8         3 5             -2 6       2 5               3 4     2 3                 -1 2   1 4                   3 0                         0 -2 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=4 }[/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}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

L11a130 L11a132