L10a86

L10a85

L10a87

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Link Presentations

[edit Notes on L10a86's Link Presentations]

Planar diagram presentation	X8192 X16,6,17,5 X18,10,19,9 X14,19,15,20 X10,16,11,15 X20,11,7,12 X4758 X2,14,3,13 X12,4,13,3 X6,18,1,17
Gauss code	{1, -8, 9, -7, 2, -10}, {7, -1, 3, -5, 6, -9, 8, -4, 5, -2, 10, -3, 4, -6}

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^2 v^4-3 u^2 v^3+4 u^2 v^2-3 u^2 v+u^2-2 u v^4+7 u v^3-11 u v^2+7 u v-2 u+v^4-3 v^3+4 v^2-3 v+1}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -9 q^{9/2}+13 q^{7/2}-\frac{1}{q^{7/2}}-17 q^{5/2}+\frac{4}{q^{5/2}}+18 q^{3/2}-\frac{9}{q^{3/2}}-q^{13/2}+4 q^{11/2}-17 \sqrt{q}+\frac{13}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +a z^5-4 z^5 a^{-1} +2 z^5 a^{-3} +2 a z^3-7 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +2 a z-5 z a^{-1} +5 z a^{-3} -z a^{-5} +a z^{-1} -2 a^{-1} z^{-1} +2 a^{-3} z^{-1} - a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -3 z^9 a^{-1} -3 z^9 a^{-3} -16 z^8 a^{-2} -8 z^8 a^{-4} -8 z^8-8 a z^7-12 z^7 a^{-1} -12 z^7 a^{-3} -8 z^7 a^{-5} -4 a^2 z^6+30 z^6 a^{-2} +11 z^6 a^{-4} -4 z^6 a^{-6} +11 z^6-a^3 z^5+14 a z^5+36 z^5 a^{-1} +36 z^5 a^{-3} +14 z^5 a^{-5} -z^5 a^{-7} +5 a^2 z^4-18 z^4 a^{-2} -4 z^4 a^{-4} +5 z^4 a^{-6} -4 z^4+a^3 z^3-9 a z^3-31 z^3 a^{-1} -31 z^3 a^{-3} -9 z^3 a^{-5} +z^3 a^{-7} -a^2 z^2+4 z^2 a^{-2} +z^2 a^{-4} -z^2 a^{-6} +z^2+4 a z+12 z a^{-1} +12 z a^{-3} +4 z a^{-5} - a^{-2} -a z^{-1} -2 a^{-1} z^{-1} -2 a^{-3} z^{-1} - a^{-5} 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 6 χ 14                     1 1 12                   3   -3 10                 6 1   5 8               7 3     -4 6             10 6       4 4           9 8         -1 2         8 9           -1 0       6 10             4 -2     3 7               -4 -4   1 6                 5 -6   3                   -3 -8 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-4 }[/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}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/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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