L11n147

L11n146

L11n148

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Link Presentations

[edit Notes on L11n147's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X22,19,7,20 X5,13,6,12 X3,10,4,11 X15,5,16,4 X11,16,12,17 X20,13,21,14 X14,21,15,22 X17,2,18,3
Gauss code	{1, 11, -6, 7, -5, -3}, {3, -1, 2, 6, -8, 5, 9, -10, -7, 8, -11, -2, 4, -9, 10, -4}

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

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/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}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/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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