L10a131

L10a130

L10a132

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Link Presentations

[edit Notes on L10a131's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X20,16,11,15 X14,8,15,7 X10,12,5,11 X8,18,9,17 X18,10,19,9 X16,20,17,19 X2536 X4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, 4, -6, 7, -5}, {5, -2, 10, -4, 3, -8, 6, -7, 8, -3}

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 t(1) t(3)^2 t(2)^2-2 t(3)^2 t(2)^2-t(1) t(3) t(2)^2+3 t(3) t(2)^2-t(2)^2-3 t(1) t(3)^2 t(2)+t(3)^2 t(2)-t(1) t(2)+4 t(1) t(3) t(2)-4 t(3) t(2)+3 t(2)+t(1) t(3)^2+2 t(1)-3 t(1) t(3)+t(3)-2}{\sqrt{t(1)} t(2) t(3)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{10}-3 q^9+6 q^8-9 q^7+11 q^6-11 q^5+11 q^4-7 q^3+6 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} -3 z^4 a^{-6} +z^4 a^{-8} +3 z^2 a^{-2} -3 z^2 a^{-4} -3 z^2 a^{-6} +2 z^2 a^{-8} +3 a^{-2} -3 a^{-4} - a^{-6} + a^{-8} + a^{-2} z^{-2} -2 a^{-4} z^{-2} + a^{-6} z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^9 a^{-5} +z^9 a^{-7} +2 z^8 a^{-4} +6 z^8 a^{-6} +4 z^8 a^{-8} +2 z^7 a^{-3} +4 z^7 a^{-5} +8 z^7 a^{-7} +6 z^7 a^{-9} +z^6 a^{-2} -z^6 a^{-4} -10 z^6 a^{-6} -3 z^6 a^{-8} +5 z^6 a^{-10} -5 z^5 a^{-3} -14 z^5 a^{-5} -22 z^5 a^{-7} -10 z^5 a^{-9} +3 z^5 a^{-11} -4 z^4 a^{-2} -10 z^4 a^{-4} -z^4 a^{-6} -2 z^4 a^{-8} -6 z^4 a^{-10} +z^4 a^{-12} +z^3 a^{-3} +8 z^3 a^{-5} +19 z^3 a^{-7} +9 z^3 a^{-9} -3 z^3 a^{-11} +6 z^2 a^{-2} +12 z^2 a^{-4} +6 z^2 a^{-6} +4 z^2 a^{-8} +3 z^2 a^{-10} -z^2 a^{-12} +4 z a^{-3} -6 z a^{-7} -2 z a^{-9} -4 a^{-2} -6 a^{-4} -3 a^{-6} - a^{-8} - a^{-10} -2 a^{-3} z^{-1} -2 a^{-5} z^{-1} + a^{-2} z^{-2} +2 a^{-4} z^{-2} + a^{-6} 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   -2 17                 4 1   3 15               6 3     -3 13             5 3       2 11           6 6         0 9         5 5           0 7       3 7             4 5     3 4               -1 3   1 5                 4 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}^{5}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=8 }[/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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