L10a144

L10a143

L10a145

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Link Presentations

[edit Notes on L10a144's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X8,18,9,17 X16,8,17,7 X18,10,19,9 X10,14,11,13 X20,12,13,11 X12,20,5,19 X2536 X4,16,1,15
Gauss code	{1, -9, 2, -10}, {9, -1, 4, -3, 5, -6, 7, -8}, {6, -2, 10, -4, 3, -5, 8, -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{t(1) t(3)^2 t(2)^3-t(3)^2 t(2)^3+t(3) t(2)^3+t(1) t(3)^3 t(2)^2-t(3)^3 t(2)^2-2 t(1) t(3)^2 t(2)^2+2 t(3)^2 t(2)^2+t(1) t(3) t(2)^2-2 t(3) t(2)^2+t(2)^2-t(1) t(3)^3 t(2)+2 t(1) t(3)^2 t(2)-t(3)^2 t(2)+t(1) t(2)-2 t(1) t(3) t(2)+2 t(3) t(2)-t(2)-t(1) t(3)^2+t(1) t(3)-t(3)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{10}-2 q^9+4 q^8-6 q^7+9 q^6-8 q^5+8 q^4-6 q^3+5 q^2-2 q+1 }[/math] (db)
Signature	4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^6 a^{-4} -z^6 a^{-6} +z^4 a^{-2} -3 z^4 a^{-4} -4 z^4 a^{-6} +z^4 a^{-8} +3 z^2 a^{-2} -z^2 a^{-4} -6 z^2 a^{-6} +3 z^2 a^{-8} +2 a^{-2} + a^{-4} -5 a^{-6} +2 a^{-8} + a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^9 a^{-5} +z^9 a^{-7} +2 z^8 a^{-4} +5 z^8 a^{-6} +3 z^8 a^{-8} +2 z^7 a^{-3} +z^7 a^{-5} +2 z^7 a^{-7} +3 z^7 a^{-9} +z^6 a^{-2} -4 z^6 a^{-4} -16 z^6 a^{-6} -8 z^6 a^{-8} +3 z^6 a^{-10} -6 z^5 a^{-3} -9 z^5 a^{-5} -10 z^5 a^{-7} -5 z^5 a^{-9} +2 z^5 a^{-11} -4 z^4 a^{-2} -z^4 a^{-4} +22 z^4 a^{-6} +12 z^4 a^{-8} -6 z^4 a^{-10} +z^4 a^{-12} +3 z^3 a^{-3} +10 z^3 a^{-5} +14 z^3 a^{-7} +4 z^3 a^{-9} -3 z^3 a^{-11} +5 z^2 a^{-2} -z^2 a^{-4} -19 z^2 a^{-6} -5 z^2 a^{-8} +6 z^2 a^{-10} -2 z^2 a^{-12} +z a^{-3} -8 z a^{-5} -8 z a^{-7} +z a^{-9} -2 a^{-2} +3 a^{-4} +9 a^{-6} +3 a^{-8} -2 a^{-10} +2 a^{-5} z^{-1} +2 a^{-7} z^{-1} - a^{-4} z^{-2} -2 a^{-6} z^{-2} - a^{-8} z^{-2} }[/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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 21                     1 1 19                   2 1 -1 17                 2     2 15               4 2     -2 13             5 2       3 11           4 5         1 9         4 4           0 7       2 4             2 5     3 4               -1 3   1 4                 3 1   1                   -1 -1 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ i=5 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=8 }[/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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