L11a126

L11a125

L11a127

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Link Presentations

[edit Notes on L11a126's Link Presentations]

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

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