L11n142

L11n141

L11n143

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Link Presentations

[edit Notes on L11n142's Link Presentations]

Planar diagram presentation	X8192 X18,11,19,12 X3,10,4,11 X17,3,18,2 X12,5,13,6 X6718 X9,16,10,17 X20,16,21,15 X22,14,7,13 X14,22,15,21 X4,20,5,19
Gauss code	{1, 4, -3, -11, 5, -6}, {6, -1, -7, 3, 2, -5, 9, -10, 8, 7, -4, -2, 11, -8, 10, -9}

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^2 v^2-2 u^2 v-3 u v^2+7 u v-3 u-2 v+2}{u v} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{5/2}-3 q^{3/2}+4 \sqrt{q}-\frac{7}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^5+2 z a^5+a^5 z^{-1} -z^5 a^3-3 z^3 a^3-4 z a^3-2 a^3 z^{-1} -z^5 a-2 z^3 a+2 a z^{-1} +z^3 a^{-1} +z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^5-3 a^7 z^3+a^7 z+2 a^6 z^6-5 a^6 z^4+a^6 z^2+3 a^5 z^7-9 a^5 z^5+8 a^5 z^3-4 a^5 z+a^5 z^{-1} +3 a^4 z^8-11 a^4 z^6+16 a^4 z^4-8 a^4 z^2+a^3 z^9-8 a^3 z^5+18 a^3 z^3-10 a^3 z+2 a^3 z^{-1} +4 a^2 z^8-16 a^2 z^6+27 a^2 z^4+z^4 a^{-2} -13 a^2 z^2-2 z^2 a^{-2} +a^2+a z^9-3 a z^7+5 a z^5+3 z^5 a^{-1} +a z^3-6 z^3 a^{-1} -4 a z+z a^{-1} +2 a z^{-1} + a^{-1} z^{-1} +z^8-3 z^6+7 z^4-6 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                 2   2 2               2 1   -1 0             5 2     3 -2           3 3       0 -4         4 4         0 -6       3 4           1 -8     1 3             -2 -10   1 3               2 -12   1                 -1 -14 1                   1

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} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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