L10a50

L10a49

L10a51

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Link Presentations

[edit Notes on L10a50's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X10,5,11,6 X14,3,15,4 X20,11,5,12 X18,13,19,14 X12,19,13,20 X2,9,3,10 X8,15,9,16
Gauss code	{1, -9, 5, -3}, {4, -1, 2, -10, 9, -4, 6, -8, 7, -5, 10, -2, 3, -7, 8, -6}

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{6 u v^2-7 u v+2 u+2 v^3-7 v^2+6 v}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{9}{q^{15/2}}+\frac{7}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{3}{q^{21/2}}-\frac{1}{q^{23/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{11} z-a^{11} z^{-1} -a^9 z^3+2 a^9 z+2 a^9 z^{-1} -3 a^7 z^3-2 a^7 z-3 a^5 z^3-3 a^5 z-a^5 z^{-1} -a^3 z^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^7 a^{13}+4 z^5 a^{13}-4 z^3 a^{13}+z a^{13}-3 z^8 a^{12}+14 z^6 a^{12}-19 z^4 a^{12}+6 z^2 a^{12}+2 a^{12}-2 z^9 a^{11}+5 z^7 a^{11}+3 z^5 a^{11}-9 z^3 a^{11}+2 z a^{11}-a^{11} z^{-1} -8 z^8 a^{10}+30 z^6 a^{10}-29 z^4 a^{10}+2 z^2 a^{10}+5 a^{10}-2 z^9 a^9-z^7 a^9+17 z^5 a^9-15 z^3 a^9+4 z a^9-2 a^9 z^{-1} -5 z^8 a^8+9 z^6 a^8+z^4 a^8-4 z^2 a^8+3 a^8-7 z^7 a^7+12 z^5 a^7-3 z^3 a^7-7 z^6 a^6+8 z^4 a^6-a^6-6 z^5 a^5+6 z^3 a^5-3 z a^5+a^5 z^{-1} -3 z^4 a^4-z^3 a^3 }[/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 -2                     1 1 -4                   3 1 -2 -6                 3     3 -8               4 3     -1 -10             6 3       3 -12           4 5         1 -14         5 5           0 -16       2 4             2 -18     2 5               -3 -20   1 2                 1 -22   2                   -2 -24 1                     1

Integral Khovanov Homology (db, data source)

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