L11n216

L11n215

L11n217

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Link Presentations

[edit Notes on L11n216's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X18,8,19,7 X14,6,15,5 X17,9,18,22 X21,17,22,16 X20,13,21,14 X6,16,7,15 X4,20,5,19 X2,9,3,10 X8,11,1,12
Gauss code	{1, -10, 2, -9, 4, -8, 3, -11}, {10, -1, 11, -2, 7, -4, 8, 6, -5, -3, 9, -7, -6, 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{(t(1)-1) (t(2)-1) \left(t(2) t(1)^2+t(2)^2 t(1)-2 t(2) t(1)+t(1)+t(2)\right)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{13/2}+3 q^{11/2}-5 q^{9/2}+7 q^{7/2}-8 q^{5/2}+8 q^{3/2}-8 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{3}{q^{3/2}}+\frac{1}{q^{5/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^{-5} -z a^{-5} +z^5 a^{-3} +2 z^3 a^{-3} +z a^{-3} +z^5 a^{-1} -a z^3+2 z^3 a^{-1} -a z+z a^{-1} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-7} -2 z^3 a^{-7} +3 z^6 a^{-6} -7 z^4 a^{-6} +2 z^2 a^{-6} +4 z^7 a^{-5} -10 z^5 a^{-5} +6 z^3 a^{-5} -2 z a^{-5} +3 z^8 a^{-4} -7 z^6 a^{-4} +6 z^4 a^{-4} -2 z^2 a^{-4} +z^9 a^{-3} +z^7 a^{-3} -6 z^5 a^{-3} +8 z^3 a^{-3} -2 z a^{-3} +4 z^8 a^{-2} -13 z^6 a^{-2} +a^2 z^4+20 z^4 a^{-2} -2 a^2 z^2-8 z^2 a^{-2} +z^9 a^{-1} -3 z^7 a^{-1} +3 a z^5+8 z^5 a^{-1} -6 a z^3-6 z^3 a^{-1} +2 a z+2 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +z^8-3 z^6+8 z^4-6 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   \ -3 -2 -1 0 1 2 3 4 5 6 χ 14                   1 1 12                 2   -2 10               3 1   2 8             4 2     -2 6           4 3       1 4         4 4         0 2       4 4           0 0     2 6             4 -2   1 2               -1 -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=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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