L11n157

L11n156

L11n158

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Link Presentations

[edit Notes on L11n157's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X17,15,18,14 X7,17,8,16 X15,7,16,22 X13,19,14,18 X6,20,1,19 X20,12,21,11 X12,6,13,5 X4,21,5,22
Gauss code	{1, -2, 3, -11, 10, -8}, {-5, -1, 2, -3, 9, -10, -7, 4, -6, 5, -4, 7, 8, -9, 11, 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{t(1)^2 t(2)^4-t(1) t(2)^4-2 t(1)^2 t(2)^3+5 t(1) t(2)^3-t(2)^3+2 t(1)^2 t(2)^2-7 t(1) t(2)^2+2 t(2)^2-t(1)^2 t(2)+5 t(1) t(2)-2 t(2)-t(1)+1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -4 q^{9/2}+7 q^{7/2}-\frac{1}{q^{7/2}}-10 q^{5/2}+\frac{3}{q^{5/2}}+10 q^{3/2}-\frac{6}{q^{3/2}}+2 q^{11/2}-11 \sqrt{q}+\frac{8}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +a z^5-5 z^5 a^{-1} +z^5 a^{-3} +3 a z^3-9 z^3 a^{-1} +2 z^3 a^{-3} +3 a z-5 z a^{-1} +z a^{-5} +a z^{-1} -2 a^{-3} z^{-1} + a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 z^4 a^{-6} -6 z^2 a^{-6} +2 a^{-6} +z^7 a^{-5} +z^5 a^{-5} -3 z^3 a^{-5} +z a^{-5} - a^{-5} z^{-1} +2 z^8 a^{-4} -4 z^6 a^{-4} +10 z^4 a^{-4} -12 z^2 a^{-4} +5 a^{-4} +z^9 a^{-3} +2 z^7 a^{-3} +a^3 z^5-5 z^5 a^{-3} -2 a^3 z^3+6 z^3 a^{-3} +a^3 z-2 a^{-3} z^{-1} +5 z^8 a^{-2} +3 a^2 z^6-9 z^6 a^{-2} -6 a^2 z^4+9 z^4 a^{-2} +2 a^2 z^2-5 z^2 a^{-2} +3 a^{-2} +z^9 a^{-1} +4 a z^7+5 z^7 a^{-1} -7 a z^5-14 z^5 a^{-1} +3 a z^3+14 z^3 a^{-1} -3 a z-5 z a^{-1} +a z^{-1} +3 z^8-2 z^6-4 z^4+3 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 χ 12                   2 -2 10                 2   2 8               5 2   -3 6             5 2     3 4           5 5       0 2         6 5         1 0       3 6           3 -2     3 5             -2 -4   1 4               3 -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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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