L10a100

L10a99

L10a101

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Link Presentations

[edit Notes on L10a100's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X14,19,15,20 X18,7,19,8 X16,5,17,6 X4,15,5,16 X6,17,7,18 X20,13,9,14 X2,9,3,10 X8,11,1,12
Gauss code	{1, -9, 2, -6, 5, -7, 4, -10}, {9, -1, 10, -2, 8, -3, 6, -5, 7, -4, 3, -8}

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{2 u^3 v^2-u^3 v+2 u^2 v^3-3 u^2 v^2+3 u^2 v-u^2-u v^3+3 u v^2-3 u v+2 u-v^2+2 v}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{4}{q^{9/2}}+\frac{2}{q^{7/2}}-\frac{1}{q^{5/2}}-\frac{1}{q^{25/2}}+\frac{2}{q^{23/2}}-\frac{4}{q^{21/2}}+\frac{6}{q^{19/2}}-\frac{7}{q^{17/2}}+\frac{8}{q^{15/2}}-\frac{8}{q^{13/2}}+\frac{5}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{11}+2 z a^{11}-z^5 a^9-2 z^3 a^9+z a^9+a^9 z^{-1} -2 z^5 a^7-7 z^3 a^7-6 z a^7-a^7 z^{-1} -z^5 a^5-3 z^3 a^5-z a^5 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{15}+3 z^3 a^{15}-2 z a^{15}-2 z^6 a^{14}+5 z^4 a^{14}-2 z^2 a^{14}-2 z^7 a^{13}+3 z^5 a^{13}+z^3 a^{13}-2 z^8 a^{12}+5 z^6 a^{12}-8 z^4 a^{12}+7 z^2 a^{12}-z^9 a^{11}+z^7 a^{11}-z^5 a^{11}-z a^{11}-4 z^8 a^{10}+13 z^6 a^{10}-19 z^4 a^{10}+7 z^2 a^{10}-z^9 a^9+6 z^5 a^9-13 z^3 a^9+5 z a^9-a^9 z^{-1} -2 z^8 a^8+4 z^6 a^8-z^4 a^8-3 z^2 a^8+a^8-3 z^7 a^7+10 z^5 a^7-12 z^3 a^7+7 z a^7-a^7 z^{-1} -2 z^6 a^6+5 z^4 a^6-z^2 a^6-z^5 a^5+3 z^3 a^5-z a^5 }[/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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                     1 1 -6                   2 1 -1 -8                 2     2 -10               3 2     -1 -12             5 2       3 -14           3 3         0 -16         4 5           -1 -18       2 3             1 -20     2 4               -2 -22   1 3                 2 -24   1                   -1 -26 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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