L10a72

L10a71

L10a73

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Link Presentations

[edit Notes on L10a72's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X14,17,15,18 X16,5,17,6 X4,15,5,16 X18,11,19,12 X20,13,7,14 X12,19,13,20 X2738 X6,9,1,10
Gauss code	{1, -9, 2, -5, 4, -10}, {9, -1, 10, -2, 6, -8, 7, -3, 5, -4, 3, -6, 8, -7}

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 t(1) t(2)^4-t(2)^4+2 t(1)^2 t(2)^3-4 t(1) t(2)^3+2 t(2)^3-2 t(1)^2 t(2)^2+5 t(1) t(2)^2-2 t(2)^2+2 t(1)^2 t(2)-4 t(1) t(2)+2 t(2)-t(1)^2+2 t(1)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{5}{q^{9/2}}+\frac{2}{q^{7/2}}-\frac{1}{q^{5/2}}-\frac{1}{q^{25/2}}+\frac{3}{q^{23/2}}-\frac{5}{q^{21/2}}+\frac{8}{q^{19/2}}-\frac{10}{q^{17/2}}+\frac{10}{q^{15/2}}-\frac{10}{q^{13/2}}+\frac{7}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{11} z^3+a^{11} z-a^{11} z^{-1} -a^9 z^5-a^9 z^3+3 a^9 z+3 a^9 z^{-1} -2 a^7 z^5-6 a^7 z^3-5 a^7 z-2 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}+2 z^3 a^{15}-3 z^6 a^{14}+7 z^4 a^{14}-3 z^2 a^{14}-4 z^7 a^{13}+9 z^5 a^{13}-6 z^3 a^{13}+2 z a^{13}-3 z^8 a^{12}+4 z^6 a^{12}-2 z^2 a^{12}+a^{12}-z^9 a^{11}-4 z^7 a^{11}+12 z^5 a^{11}-13 z^3 a^{11}+5 z a^{11}-a^{11} z^{-1} -5 z^8 a^{10}+9 z^6 a^{10}-5 z^4 a^{10}-5 z^2 a^{10}+3 a^{10}-z^9 a^9-3 z^7 a^9+10 z^5 a^9-16 z^3 a^9+12 z a^9-3 a^9 z^{-1} -2 z^8 a^8+6 z^4 a^8-7 z^2 a^8+3 a^8-3 z^7 a^7+7 z^5 a^7-8 z^3 a^7+7 z a^7-2 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             6 3       3 -14           4 4         0 -16         6 6           0 -18       3 5             2 -20     2 5               -3 -22   1 3                 2 -24   2                   -2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\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^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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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