L11a225

L11a224

L11a226

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Link Presentations

[edit Notes on L11a225's Link Presentations]

Planar diagram presentation	X8192 X16,6,17,5 X18,10,19,9 X10,20,11,19 X2,11,3,12 X12,3,13,4 X4758 X20,16,21,15 X22,14,7,13 X14,22,15,21 X6,18,1,17
Gauss code	{1, -5, 6, -7, 2, -11}, {7, -1, 3, -4, 5, -6, 9, -10, 8, -2, 11, -3, 4, -8, 10, -9}

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 u^2 v^4-4 u^2 v^3+4 u^2 v^2-2 u^2 v-2 u v^4+8 u v^3-11 u v^2+8 u v-2 u-2 v^3+4 v^2-4 v+2}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{15/2}-3 q^{13/2}+7 q^{11/2}-11 q^{9/2}+15 q^{7/2}-18 q^{5/2}+17 q^{3/2}-16 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{7/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^{-5} +3 z^3 a^{-5} +3 z a^{-5} + a^{-5} z^{-1} -z^7 a^{-3} -4 z^5 a^{-3} -6 z^3 a^{-3} -4 z a^{-3} -2 a^{-3} z^{-1} -z^7 a^{-1} +a z^5-4 z^5 a^{-1} +3 a z^3-6 z^3 a^{-1} +3 a z-3 z a^{-1} +a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-8} -3 z^4 a^{-8} +2 z^2 a^{-8} +3 z^7 a^{-7} -8 z^5 a^{-7} +4 z^3 a^{-7} +5 z^8 a^{-6} -14 z^6 a^{-6} +13 z^4 a^{-6} -9 z^2 a^{-6} +2 a^{-6} +5 z^9 a^{-5} -13 z^7 a^{-5} +14 z^5 a^{-5} -12 z^3 a^{-5} +4 z a^{-5} - a^{-5} z^{-1} +2 z^{10} a^{-4} +4 z^8 a^{-4} -24 z^6 a^{-4} +35 z^4 a^{-4} -22 z^2 a^{-4} +5 a^{-4} +10 z^9 a^{-3} -28 z^7 a^{-3} +a^3 z^5+32 z^5 a^{-3} -2 a^3 z^3-14 z^3 a^{-3} +a^3 z+5 z a^{-3} -2 a^{-3} z^{-1} +2 z^{10} a^{-2} +5 z^8 a^{-2} +3 a^2 z^6-22 z^6 a^{-2} -5 a^2 z^4+30 z^4 a^{-2} +a^2 z^2-13 z^2 a^{-2} +3 a^{-2} +5 z^9 a^{-1} +5 a z^7-7 z^7 a^{-1} -8 a z^5+z^5 a^{-1} +4 a z^3+8 z^3 a^{-1} -4 a z-4 z a^{-1} +a z^{-1} +6 z^8-10 z^6+6 z^4-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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     2   2 12                   5 1   -4 10                 6 2     4 8               9 5       -4 6             9 6         3 4           8 9           1 2         8 9             -1 0       4 9               5 -2     3 7                 -4 -4   1 5                   4 -6   2                     -2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\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^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/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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