L11n132

L11n131

L11n133

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Link Presentations

[edit Notes on L11n132'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 X13,20,14,21 X15,22,16,7 X4,20,5,19 X21,14,22,15
Gauss code	{1, 4, -3, -10, 5, -6}, {6, -1, -7, 3, 2, -5, -8, 11, -9, 7, -4, -2, 10, 8, -11, 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{t(1) t(2)^4-t(1)^2 t(2)^3+t(1)^2 t(2)^2-t(1) t(2)^2+t(2)^2-t(2)+t(1)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{21/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{15/2}}+\frac{1}{q^{13/2}}-\frac{1}{q^{11/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^9-3 z a^9-2 a^9 z^{-1} +z^5 a^7+6 z^3 a^7+11 z a^7+5 a^7 z^{-1} -z^5 a^5-6 z^3 a^5-9 z a^5-3 a^5 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^{12}+5 z^4 a^{12}-6 z^2 a^{12}+a^{12}-z^7 a^{11}+5 z^5 a^{11}-6 z^3 a^{11}+z a^{11}-z^6 a^{10}+4 z^4 a^{10}-3 z^2 a^{10}-z^5 a^9+4 z^3 a^9-4 z a^9+2 a^9 z^{-1} +z^6 a^8-7 z^4 a^8+13 z^2 a^8-5 a^8+z^7 a^7-7 z^5 a^7+16 z^3 a^7-15 z a^7+5 a^7 z^{-1} +z^6 a^6-6 z^4 a^6+10 z^2 a^6-5 a^6-z^5 a^5+6 z^3 a^5-10 z a^5+3 a^5 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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                   1 1 -6               1   1 2 -8             1 2     1 -10           1         1 -12         1 2 1       0 -14       1 1           0 -16       1 1           0 -18   1 1               0 -20                     0 -22 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{ i=-2 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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