L11n148

L11n147

L11n149

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Link Presentations

[edit Notes on L11n148's Link Presentations]

Planar diagram presentation	X8192 X9,19,10,18 X6718 X19,7,20,22 X12,5,13,6 X3,10,4,11 X4,15,5,16 X16,12,17,11 X13,21,14,20 X21,15,22,14 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{t(1) t(2)^4+t(2)^4-t(1) t(2)^3+t(1) t(2)^2-t(1) t(2)+t(1)^2+t(1)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{5/2}-\frac{1}{q^{5/2}}-q^{3/2}+\frac{1}{q^{3/2}}-\frac{1}{q^{11/2}}+\sqrt{q}-\frac{2}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^5+a^5 z^{-1} -z^5 a-5 z^3 a-7 z a-2 a z^{-1} +z^3 a^{-1} +3 z a^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^5 z^7+a z^7+a^2 z^6+z^6+7 a^5 z^5+a^3 z^5-7 a z^5-z^5 a^{-1} +a^4 z^4-6 a^2 z^4-z^4 a^{-2} -8 z^4-14 a^5 z^3-4 a^3 z^3+13 a z^3+3 z^3 a^{-1} -3 a^4 z^2+8 a^2 z^2+3 z^2 a^{-2} +14 z^2+8 a^5 z+2 a^3 z-9 a z-3 z a^{-1} +a^4-3 a^2-2 a^{-2} -5-a^5 z^{-1} +2 a z^{-1} + a^{-1} 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 χ 6                   1 -1 4                     0 2               1 1   0 0           1 2       1 -2           1 2       1 -4       1 2 1         0 -6         1           1 -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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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.

L11n147 L11n149