L10a116

L10a115

L10a117

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Symmetric depiction

Link Presentations

[edit Notes on L10a116's Link Presentations]

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

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(2)^2 t(1)^4-2 t(2)^3 t(1)^3+3 t(2)^2 t(1)^3-2 t(2) t(1)^3-t(2)^4 t(1)^2+3 t(2)^3 t(1)^2-5 t(2)^2 t(1)^2+3 t(2) t(1)^2-t(1)^2-2 t(2)^3 t(1)+3 t(2)^2 t(1)-2 t(2) t(1)-t(2)^2}{t(1)^2 t(2)^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{10}{q^{15/2}}-\frac{9}{q^{17/2}}+\frac{7}{q^{19/2}}-\frac{5}{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^{11} z^{-1} -a^9 z^5-2 a^9 z^3-a^9 z-a^9 z^{-1} -2 a^7 z^5-6 a^7 z^3-4 a^7 z-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}+4 z^4 a^{14}-z^2 a^{14}-3 z^7 a^{13}+7 z^5 a^{13}-7 z^3 a^{13}+4 z a^{13}-2 z^8 a^{12}+z^6 a^{12}+2 z^4 a^{12}-z^2 a^{12}-z^9 a^{11}-z^7 a^{11}+3 z^3 a^{11}-3 z a^{11}+a^{11} z^{-1} -4 z^8 a^{10}+6 z^6 a^{10}-4 z^4 a^{10}-a^{10}-z^9 a^9-z^7 a^9+3 z^3 a^9-3 z a^9+a^9 z^{-1} -2 z^8 a^8+z^6 a^8+2 z^4 a^8-z^2 a^8-3 z^7 a^7+7 z^5 a^7-7 z^3 a^7+4 z a^7-2 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           5 4         -1 -16         4 5           -1 -18       3 5             2 -20     2 4               -2 -22     3                 3 -24 1 2                   -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{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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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