L11n454

L11n453

L11n455

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Link Presentations

[edit Notes on L11n454's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X2,16,3,15 X16,7,17,8 X19,22,20,15 X21,14,22,11 X13,20,14,21 X9,18,10,19 X11,10,12,5 X17,1,18,4
Gauss code	{1, -4, -3, 11}, {-10, 2, -8, 7}, {-2, -1, 5, 3, -9, 10}, {4, -5, -11, 9, -6, 8, -7, 6}

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{2 u v w x^2-2 u v w x-u v x^2+u v x-u w x^2+2 u w x+u x^2-2 u x-2 v w^2 x+v w^2+2 v w x-v w+w^2 x-w^2-2 w x+2 w}{\sqrt{u} \sqrt{v} w x} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{6}{q^{9/2}}-\frac{7}{q^{7/2}}+\frac{2}{q^{5/2}}-\frac{2}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{7}{q^{15/2}}+\frac{7}{q^{13/2}}-\frac{9}{q^{11/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 \left(-z^3\right)+a^9 z^{-3} -a^9 z+a^9 z^{-1} +a^7 z^5+2 a^7 z^3-3 a^7 z^{-3} -a^7 z-6 a^7 z^{-1} +a^5 z^5+3 a^5 z^3+3 a^5 z^{-3} +7 a^5 z+9 a^5 z^{-1} -2 a^3 z^3-a^3 z^{-3} -5 a^3 z-4 a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^{12}+3 z^4 a^{12}-z^2 a^{12}-3 z^7 a^{11}+11 z^5 a^{11}-10 z^3 a^{11}+3 z a^{11}-3 z^8 a^{10}+9 z^6 a^{10}-4 z^4 a^{10}-z^2 a^{10}-z^9 a^9-3 z^7 a^9+21 z^5 a^9-28 z^3 a^9+16 z a^9-5 a^9 z^{-1} +a^9 z^{-3} -5 z^8 a^8+15 z^6 a^8-10 z^4 a^8-7 z^2 a^8-3 a^8 z^{-2} +10 a^8-z^9 a^7-2 z^7 a^7+16 z^5 a^7-31 z^3 a^7+25 z a^7-12 a^7 z^{-1} +3 a^7 z^{-3} -2 z^8 a^6+4 z^6 a^6-z^4 a^6-15 z^2 a^6-6 a^6 z^{-2} +19 a^6-2 z^7 a^5+6 z^5 a^5-16 z^3 a^5+19 z a^5-12 a^5 z^{-1} +3 a^5 z^{-3} -z^6 a^4+2 z^4 a^4-8 z^2 a^4-3 a^4 z^{-2} +10 a^4-3 z^3 a^3+7 z a^3-5 a^3 z^{-1} +a^3 z^{-3} }[/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 -2                   2 2 -4                 2 2 0 -6               5     5 -8             3 4     1 -10           6 3       3 -12         3 5         2 -14       4 4           0 -16     2 5             3 -18   1 2               -1 -20   2                 2 -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=-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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/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}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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