L11a153

L11a152

L11a154

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Link Presentations

[edit Notes on L11a153's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X20,16,21,15 X14,5,15,6 X4,13,5,14 X22,18,7,17 X16,22,17,21 X12,20,13,19 X18,12,19,11 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 9, -8, 5, -4, 3, -7, 6, -9, 8, -3, 7, -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^5-2 u^2 v^4+3 u^2 v^3-3 u^2 v^2+2 u^2 v+u v^6-3 u v^5+6 u v^4-7 u v^3+6 u v^2-3 u v+u+2 v^5-3 v^4+3 v^3-2 v^2+v}{u v^3} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -3 q^{9/2}+\frac{2}{q^{9/2}}+7 q^{7/2}-\frac{6}{q^{7/2}}-11 q^{5/2}+\frac{9}{q^{5/2}}+14 q^{3/2}-\frac{13}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-16 \sqrt{q}+\frac{15}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a z^7-z^7 a^{-1} +a^3 z^5-5 a z^5-4 z^5 a^{-1} +z^5 a^{-3} +4 a^3 z^3-11 a z^3-5 z^3 a^{-1} +3 z^3 a^{-3} +6 a^3 z-12 a z+3 z a^{-3} +3 a^3 z^{-1} -5 a z^{-1} +2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^2 z^{10}-z^{10}-3 a^3 z^9-7 a z^9-4 z^9 a^{-1} -2 a^4 z^8-5 a^2 z^8-7 z^8 a^{-2} -10 z^8-a^5 z^7+11 a^3 z^7+19 a z^7-z^7 a^{-1} -8 z^7 a^{-3} +7 a^4 z^6+26 a^2 z^6+8 z^6 a^{-2} -6 z^6 a^{-4} +33 z^6+5 a^5 z^5-16 a^3 z^5-22 a z^5+14 z^5 a^{-1} +12 z^5 a^{-3} -3 z^5 a^{-5} -6 a^4 z^4-33 a^2 z^4+7 z^4 a^{-4} -z^4 a^{-6} -35 z^4-8 a^5 z^3+17 a^3 z^3+29 a z^3-8 z^3 a^{-1} -10 z^3 a^{-3} +2 z^3 a^{-5} +19 a^2 z^2-3 z^2 a^{-2} -5 z^2 a^{-4} +z^2 a^{-6} +22 z^2+4 a^5 z-12 a^3 z-21 a z-2 z a^{-1} +3 z a^{-3} -5 a^2+ a^{-4} -5+3 a^3 z^{-1} +5 a z^{-1} +2 a^{-1} 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     2   2 8                   5 1   -4 6                 6 2     4 4               8 5       -3 2             8 6         2 0           8 9           1 -2         5 7             -2 -4       4 8               4 -6     2 5                 -3 -8     4                   4 -10 1 2                     -1 -12 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/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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