L11n77

L11n76

L11n78

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Link Presentations

[edit Notes on L11n77's Link Presentations]

Planar diagram presentation	X6172 X3,10,4,11 X7,20,8,21 X15,5,16,22 X21,17,22,16 X9,14,10,15 X13,19,14,18 X19,13,20,12 X17,8,18,9 X2536 X11,4,12,1
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 9, -6, 2, -11, 8, -7, 6, -4, 5, -9, 7, -8, 3, -5, 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 v^5-4 u v^4+7 u v^3-3 u v^2-3 v^3+7 v^2-4 v+1}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{5/2}+3 q^{3/2}-6 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{9}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{1}{q^{13/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^7+z^5 a^5+4 z^3 a^5+5 z a^5+2 a^5 z^{-1} -z^7 a^3-5 z^5 a^3-10 z^3 a^3-11 z a^3-4 a^3 z^{-1} +2 z^5 a+7 z^3 a+7 z a+3 a z^{-1} -z^3 a^{-1} -2 z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^2+5 a^7 z^3-a^7 z+3 a^6 z^6-2 a^6 z^4+a^6+8 a^5 z^7-26 a^5 z^5+31 a^5 z^3-14 a^5 z+2 a^5 z^{-1} +7 a^4 z^8-22 a^4 z^6+22 a^4 z^4-12 a^4 z^2+2 a^4+2 a^3 z^9+6 a^3 z^7-42 a^3 z^5+53 a^3 z^3-26 a^3 z+4 a^3 z^{-1} +10 a^2 z^8-37 a^2 z^6+38 a^2 z^4-16 a^2 z^2+3 a^2+2 a z^9-a z^7+z^7 a^{-1} -20 a z^5-4 z^5 a^{-1} +33 a z^3+6 z^3 a^{-1} -17 a z-4 z a^{-1} +3 a z^{-1} + a^{-1} z^{-1} +3 z^8-12 z^6+14 z^4-5 z^2+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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                   1 1 4                 2   -2 2               4 1   3 0             4 2     -2 -2           6 4       2 -4         5 5         0 -6       4 5           -1 -8     3 5             2 -10   2 4               -2 -12   4                 4 -14 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/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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