L10a94

L10a93

L10a95

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Link Presentations

[edit Notes on L10a94's Link Presentations]

Planar diagram presentation	X10,1,11,2 X14,5,15,6 X12,3,13,4 X4,13,5,14 X16,19,17,20 X18,7,19,8 X6,17,7,18 X20,15,9,16 X2,9,3,10 X8,11,1,12
Gauss code	{1, -9, 3, -4, 2, -7, 6, -10}, {9, -1, 10, -3, 4, -2, 8, -5, 7, -6, 5, -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{(t(1) t(2)+1) \left(2 t(2) t(1)^2-t(1)^2+2 t(2)^2 t(1)-4 t(2) t(1)+2 t(1)-t(2)^2+2 t(2)\right)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{9}{q^{13/2}}+\frac{9}{q^{15/2}}-\frac{9}{q^{17/2}}+\frac{7}{q^{19/2}}-\frac{4}{q^{21/2}}+\frac{2}{q^{23/2}}-\frac{1}{q^{25/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{11} z^3+2 a^{11} z-a^9 z^5-2 a^9 z^3+a^9 z^{-1} -2 a^7 z^5-6 a^7 z^3-4 a^7 z-a^7 z^{-1} -a^5 z^5-3 a^5 z^3-2 a^5 z }[/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}-3 z^2 a^{14}-2 z^7 a^{13}+2 z^5 a^{13}+2 z^3 a^{13}-z a^{13}-2 z^8 a^{12}+3 z^6 a^{12}-4 z^4 a^{12}+3 z^2 a^{12}-z^9 a^{11}-3 z^5 a^{11}+6 z^3 a^{11}-3 z a^{11}-4 z^8 a^{10}+8 z^6 a^{10}-10 z^4 a^{10}+4 z^2 a^{10}-z^9 a^9-z^7 a^9+2 z^5 a^9-3 z^3 a^9+3 z a^9-a^9 z^{-1} -2 z^8 a^8+z^6 a^8+3 z^4 a^8-3 z^2 a^8+a^8-3 z^7 a^7+7 z^5 a^7-7 z^3 a^7+5 z a^7-a^7 z^{-1} -2 z^6 a^6+4 z^4 a^6-z^2 a^6-z^5 a^5+3 z^3 a^5-2 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                 3     3 -10               4 2     -2 -12             5 3       2 -14           4 4         0 -16         5 5           0 -18       2 4             2 -20     2 5               -3 -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}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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