L11a287

L11a286

L11a288

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Link Presentations

[edit Notes on L11a287's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X16,9,17,10 X20,12,21,11 X22,15,9,16 X14,6,15,5 X18,8,19,7 X4,14,5,13 X6,18,7,17 X8,20,1,19 X2,21,3,22
Gauss code	{1, -11, 2, -8, 6, -9, 7, -10}, {3, -1, 4, -2, 8, -6, 5, -3, 9, -7, 10, -4, 11, -5}

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+1) \left(u v^2-u v+u+v-1\right) \left(u v^2-u v-v^2+v-1\right)}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 14 q^{9/2}-16 q^{7/2}+15 q^{5/2}-\frac{1}{q^{5/2}}-14 q^{3/2}+\frac{3}{q^{3/2}}+q^{17/2}-3 q^{15/2}+6 q^{13/2}-11 q^{11/2}+10 \sqrt{q}-\frac{6}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^7 a^{-5} +5 z^5 a^{-5} +8 z^3 a^{-5} +3 z a^{-5} - a^{-5} z^{-1} -z^9 a^{-3} -7 z^7 a^{-3} -18 z^5 a^{-3} -19 z^3 a^{-3} -5 z a^{-3} + a^{-3} z^{-1} +z^7 a^{-1} +5 z^5 a^{-1} +8 z^3 a^{-1} +4 z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 z^{10} a^{-2} -2 z^{10} a^{-4} -4 z^9 a^{-1} -9 z^9 a^{-3} -5 z^9 a^{-5} +z^8 a^{-2} -3 z^8 a^{-4} -7 z^8 a^{-6} -3 z^8-a z^7+15 z^7 a^{-1} +29 z^7 a^{-3} +6 z^7 a^{-5} -7 z^7 a^{-7} +11 z^6 a^{-2} +14 z^6 a^{-4} +10 z^6 a^{-6} -5 z^6 a^{-8} +12 z^6+4 a z^5-19 z^5 a^{-1} -35 z^5 a^{-3} +9 z^5 a^{-7} -3 z^5 a^{-9} -12 z^4 a^{-2} -7 z^4 a^{-4} -4 z^4 a^{-6} +4 z^4 a^{-8} -z^4 a^{-10} -14 z^4-4 a z^3+13 z^3 a^{-1} +26 z^3 a^{-3} +z^3 a^{-5} -5 z^3 a^{-7} +3 z^3 a^{-9} +4 z^2 a^{-2} -2 z^2 a^{-6} -z^2 a^{-8} +z^2 a^{-10} +4 z^2-5 z a^{-1} -5 z a^{-3} -z a^{-7} -z a^{-9} + a^{-4} - 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 7 χ 18                       1 -1 16                     2   2 14                   4 1   -3 12                 7 2     5 10               8 5       -3 8             8 6         2 6           7 8           1 4         7 8             -1 2       4 8               4 0     2 6                 -4 -2   1 4                   3 -4   2                     -2 -6 1                       1

Integral Khovanov Homology (db, data source)

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