L10a146

L10a145

L10a147

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Link Presentations

[edit Notes on L10a146's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X20,12,13,11 X18,8,19,7 X16,10,17,9 X8,18,9,17 X10,14,11,13 X12,20,5,19 X2536 X4,16,1,15
Gauss code	{1, -9, 2, -10}, {9, -1, 4, -6, 5, -7, 3, -8}, {7, -2, 10, -5, 6, -4, 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{u v^3 w+2 u v^2 w^2-3 u v^2 w+u v^2+u v w^3-3 u v w^2+2 u v w-u w^3+u w^2-v^3 w+v^3-2 v^2 w^2+3 v^2 w-v^2-v w^3+3 v w^2-2 v w-w^2}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^9-2 q^8+5 q^7-7 q^6+10 q^5-9 q^4+10 q^3-8 q^2+5 q+ q^{-1} -2 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^{-2} -2 z^4 a^{-4} -z^4 a^{-6} -2 z^2 a^{-4} -z^2 a^{-6} +z^2 a^{-8} +z^2+ a^{-4} -3 a^{-6} + a^{-8} +1+ a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-10} -4 z^4 a^{-10} +4 z^2 a^{-10} +2 z^7 a^{-9} -6 z^5 a^{-9} +4 z^3 a^{-9} +3 z^8 a^{-8} -11 z^6 a^{-8} +18 z^4 a^{-8} -19 z^2 a^{-8} - a^{-8} z^{-2} +8 a^{-8} +z^9 a^{-7} +3 z^7 a^{-7} -15 z^5 a^{-7} +17 z^3 a^{-7} -11 z a^{-7} +2 a^{-7} z^{-1} +6 z^8 a^{-6} -18 z^6 a^{-6} +28 z^4 a^{-6} -30 z^2 a^{-6} -2 a^{-6} z^{-2} +13 a^{-6} +z^9 a^{-5} +5 z^7 a^{-5} -17 z^5 a^{-5} +21 z^3 a^{-5} -11 z a^{-5} +2 a^{-5} z^{-1} +3 z^8 a^{-4} -3 z^6 a^{-4} +3 z^4 a^{-4} -5 z^2 a^{-4} - a^{-4} z^{-2} +5 a^{-4} +4 z^7 a^{-3} -6 z^5 a^{-3} +6 z^3 a^{-3} +3 z^6 a^{-2} -2 z^4 a^{-2} +2 z^5 a^{-1} -2 z^3 a^{-1} +z^4-2 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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 19                     1 1 17                   2 1 -1 15                 3     3 13               4 2     -2 11             6 3       3 9           4 5         1 7         6 5           1 5       2 4             2 3     3 6               -3 1   1 4                 3 -1   1                   -1 -3 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/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}^{6}\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^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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