L11n189

L11n188

L11n190

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Link Presentations

[edit Notes on L11n189's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X22,10,7,9 X10,14,11,13 X5,16,6,17 X15,21,16,20 X21,19,22,18 X19,15,20,14 X2738 X4,11,5,12 X17,6,18,1
Gauss code	{1, -9, 2, -10, -5, 11}, {9, -1, 3, -4, 10, -2, 4, 8, -6, 5, -11, 7, -8, 6, -7, -3}

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{u^2 v^3-2 u^2 v^2+u^2 v+u^2-2 u v^3+5 u v^2-2 u v+v^4+v^3-2 v^2+v}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-3 q^{7/2}+4 q^{5/2}-\frac{3}{q^{5/2}}-5 q^{3/2}+\frac{4}{q^{3/2}}-\frac{1}{q^{11/2}}+6 \sqrt{q}-\frac{5}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^5+2 a^5 z^{-1} -z^3 a^3-5 z a^3-3 a^3 z^{-1} +2 z^3 a+3 z a+a z^{-1} -z^5 a^{-1} -3 z^3 a^{-1} -3 z a^{-1} +z^3 a^{-3} +z a^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a z^9-z^9 a^{-1} -a^2 z^8-3 z^8 a^{-2} -4 z^8-a^5 z^7+4 a z^7-3 z^7 a^{-3} +6 a^2 z^6+11 z^6 a^{-2} -z^6 a^{-4} +18 z^6+7 a^5 z^5+3 a^3 z^5-4 a z^5+11 z^5 a^{-1} +11 z^5 a^{-3} +3 a^4 z^4-10 a^2 z^4-9 z^4 a^{-2} +3 z^4 a^{-4} -25 z^4-14 a^5 z^3-11 a^3 z^3-a z^3-12 z^3 a^{-1} -8 z^3 a^{-3} -7 a^4 z^2+3 z^2 a^{-2} -z^2 a^{-4} +11 z^2+10 a^5 z+10 a^3 z+a z+2 z a^{-1} +z a^{-3} +3 a^4+3 a^2+1-2 a^5 z^{-1} -3 a^3 z^{-1} -a z^{-1} }[/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 4 5 χ 10                       1 -1 8                     2   2 6                   2 1   -1 4                 3 2     1 2               3 2       -1 0           1 3 3         -1 -2           3 4           1 -4       1 2 2             -1 -6         3               3 -8     1 1                 0 -10 1                       1 -12 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{ i=2 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/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}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/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}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/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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