L11n79

L11n78

L11n80

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Link Presentations

[edit Notes on L11n79's Link Presentations]

Planar diagram presentation	X6172 X16,9,17,10 X4,21,1,22 X11,14,12,15 X3,10,4,11 X5,13,6,12 X13,5,14,22 X15,2,16,3 X20,18,21,17 X18,8,19,7 X8,20,9,19
Gauss code	{1, 8, -5, -3}, {-6, -1, 10, -11, 2, 5, -4, 6, -7, 4, -8, -2, 9, -10, 11, -9, 3, 7}

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+6 u v^3-4 u v^2-4 v^3+6 v^2-4 v+1}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-\frac{2}{q^{9/2}}-3 q^{7/2}+\frac{4}{q^{7/2}}+6 q^{5/2}-\frac{7}{q^{5/2}}-8 q^{3/2}+\frac{9}{q^{3/2}}+9 \sqrt{q}-\frac{11}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z+a^5 z^{-1} -a^3 z^5-4 a^3 z^3+z^3 a^{-3} -5 a^3 z+2 z a^{-3} -2 a^3 z^{-1} + a^{-3} z^{-1} +a z^7+5 a z^5-2 z^5 a^{-1} +9 a z^3-7 z^3 a^{-1} +7 a z-7 z a^{-1} +3 a z^{-1} -3 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a z^9-z^9 a^{-1} -3 a^2 z^8-3 z^8 a^{-2} -6 z^8-3 a^3 z^7-6 a z^7-6 z^7 a^{-1} -3 z^7 a^{-3} -a^4 z^6+6 a^2 z^6+5 z^6 a^{-2} -z^6 a^{-4} +13 z^6+7 a^3 z^5+24 a z^5+26 z^5 a^{-1} +9 z^5 a^{-3} -2 a^4 z^4-10 a^2 z^4+6 z^4 a^{-2} +3 z^4 a^{-4} -5 z^4-3 a^5 z^3-15 a^3 z^3-32 a z^3-27 z^3 a^{-1} -7 z^3 a^{-3} +2 a^4 z^2+7 a^2 z^2-8 z^2 a^{-2} -3 z^2 a^{-4} +4 a^5 z+10 a^3 z+16 a z+13 z a^{-1} +3 z a^{-3} -2 a^2+2 a^{-2} + a^{-4} -a^5 z^{-1} -2 a^3 z^{-1} -3 a z^{-1} -3 a^{-1} z^{-1} - 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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 2   2 6               4 1   -3 4             4 2     2 2           5 4       -1 0         6 4         2 -2       4 6           2 -4     3 5             -2 -6   1 4               3 -8 1 3                 -2 -10 2                   2

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{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/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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