L11a397

L11a396

L11a398

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Link Presentations

[edit Notes on L11a397's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X22,12,9,11 X20,14,21,13 X18,16,19,15 X8,18,5,17 X16,8,17,7 X14,20,15,19 X12,22,13,21 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 7, -6}, {11, -2, 3, -9, 4, -8, 5, -7, 6, -5, 8, -4, 9, -3}

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{-3 t(2) t(1)+3 t(2) t(3) t(1)-3 t(3) t(1)+4 t(1)+3 t(2)-4 t(2) t(3)+3 t(3)-3}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^7+3 q^6-4 q^5+5 q^4-7 q^3+8 q^2-7 q+7-4 q^{-1} +4 q^{-2} - q^{-3} + q^{-4} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^{-6} +z^4 a^{-4} +a^4 z^{-2} +z^2 a^{-4} +a^4+z^4 a^{-2} -2 a^2 z^2-2 a^2 z^{-2} -3 a^2+z^4+z^2+ z^{-2} +2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^7 a^{-7} -4 z^5 a^{-7} +3 z^3 a^{-7} +3 z^8 a^{-6} -14 z^6 a^{-6} +16 z^4 a^{-6} -3 z^2 a^{-6} +3 z^9 a^{-5} -14 z^7 a^{-5} +18 z^5 a^{-5} -7 z^3 a^{-5} +z^{10} a^{-4} -z^8 a^{-4} -8 z^6 a^{-4} +a^4 z^4+10 z^4 a^{-4} -3 a^4 z^2-2 z^2 a^{-4} -a^4 z^{-2} +3 a^4+4 z^9 a^{-3} -17 z^7 a^{-3} +a^3 z^5+22 z^5 a^{-3} -10 z^3 a^{-3} -3 a^3 z+2 a^3 z^{-1} +z^{10} a^{-2} -3 z^8 a^{-2} +a^2 z^6+5 z^6 a^{-2} +2 a^2 z^4-6 z^4 a^{-2} -6 a^2 z^2+z^2 a^{-2} -2 a^2 z^{-2} +5 a^2+z^9 a^{-1} +a z^7-z^7 a^{-1} +a z^5-3 a z+2 a z^{-1} +z^8+z^4-3 z^2- z^{-2} +3 }[/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 15                       1 -1 13                     2   2 11                   2 1   -1 9                 3 2     1 7               4 2       -2 5             4 3         1 3           3 4           1 1         4 4             0 -1       3 6               3 -3     1 1                 0 -5     3                   3 -7 1 1                     0 -9 1                       1

Integral Khovanov Homology (db, data source)

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