L11n101

L11n100

L11n102

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Link Presentations

[edit Notes on L11n101's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,21,14,20 X16,7,17,8 X8,19,9,20 X18,9,19,10 X10,17,11,18 X15,5,16,22 X21,15,22,14 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 6, -7, 11, -2, -3, 9, -8, -4, 7, -6, 5, 3, -9, 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 v^3-5 u v^2+7 u v-3 u-3 v^3+7 v^2-5 v+2}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{5/2}-4 q^{3/2}+7 \sqrt{q}-\frac{10}{\sqrt{q}}+\frac{11}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{2}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^{-1} +z^3 a^5-z a^5-2 a^5 z^{-1} -z^5 a^3-z^3 a^3+2 a^3 z^{-1} -z^5 a-z^3 a-z a-a z^{-1} +z^3 a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 a^7 z^5-8 a^7 z^3+5 a^7 z-a^7 z^{-1} +a^6 z^8-4 a^6 z^4+3 a^6 z^2+a^5 z^9+2 a^5 z^5-11 a^5 z^3+9 a^5 z-2 a^5 z^{-1} +5 a^4 z^8-7 a^4 z^6-a^4 z^4+4 a^4 z^2-a^4+a^3 z^9+7 a^3 z^7-13 a^3 z^5+a^3 z^3+6 a^3 z-2 a^3 z^{-1} +4 a^2 z^8-6 a^2 z^4+z^4 a^{-2} +2 a^2 z^2+7 a z^7-8 a z^5+4 z^5 a^{-1} +a z^3-3 z^3 a^{-1} +2 a z-a z^{-1} +7 z^6-8 z^4+z^2 }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                   1 -1 4                 3   3 2               4 1   -3 0             6 3     3 -2           6 5       -1 -4         6 5         1 -6       4 6           2 -8     3 6             -3 -10   2 5               3 -12   2                 -2 -14 2                   2

Integral Khovanov Homology (db, data source)

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