L11n160

L11n159

L11n161

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Link Presentations

[edit Notes on L11n160's Link Presentations]

Planar diagram presentation	X8192 X16,7,17,8 X10,4,11,3 X2,15,3,16 X14,10,15,9 X11,19,12,18 X5,13,6,12 X6,21,1,22 X20,14,21,13 X22,17,7,18 X19,4,20,5
Gauss code	{1, -4, 3, 11, -7, -8}, {2, -1, 5, -3, -6, 7, 9, -5, 4, -2, 10, 6, -11, -9, 8, -10}

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 v-u-v+2) (2 u v-u-v+1)}{u v} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-3 q^{7/2}+5 q^{5/2}-7 q^{3/2}+8 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{1}{q^{9/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^3+z^3 a^{-3} +a^3 z+z a^{-3} +a^3 z^{-1} -a z^5-z^5 a^{-1} -2 a z^3-2 z^3 a^{-1} -2 a z-z a^{-1} -a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a z^9-2 z^9 a^{-1} -3 a^2 z^8-4 z^8 a^{-2} -7 z^8-a^3 z^7+5 a z^7+3 z^7 a^{-1} -3 z^7 a^{-3} +11 a^2 z^6+14 z^6 a^{-2} -z^6 a^{-4} +26 z^6-3 a z^5+7 z^5 a^{-1} +10 z^5 a^{-3} -3 a^4 z^4-18 a^2 z^4-12 z^4 a^{-2} +3 z^4 a^{-4} -30 z^4-a^5 z^3-a^3 z^3-3 a z^3-10 z^3 a^{-1} -7 z^3 a^{-3} +3 a^4 z^2+8 a^2 z^2+4 z^2 a^{-2} -z^2 a^{-4} +10 z^2+a^5 z+3 z a^{-1} +2 z a^{-3} -a^2+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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 2   2 6               3 1   -2 4             4 2     2 2           4 3       -1 0         5 4         1 -2       3 5           2 -4     3 4             -1 -6   1 4               3 -8   2                 -2 -10 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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/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}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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