L11a313

L11a312

L11a314

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Link Presentations

[edit Notes on L11a313's Link Presentations]

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

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) (t(1)+t(2)) (t(1) t(2)+1) \left(t(2)^2-t(2)+1\right)}{t(1)^{3/2} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{11/2}+3 q^{9/2}-6 q^{7/2}+10 q^{5/2}-13 q^{3/2}+14 \sqrt{q}-\frac{16}{\sqrt{q}}+\frac{13}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{1}{q^{11/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^3 z^5-z^5 a^{-3} -3 a^3 z^3-3 z^3 a^{-3} -2 a^3 z-2 z a^{-3} +a z^7+z^7 a^{-1} +4 a z^5+4 z^5 a^{-1} +5 a z^3+5 z^3 a^{-1} +2 a z+2 z a^{-1} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 z^{10} a^{-2} -2 z^{10}-5 a z^9-9 z^9 a^{-1} -4 z^9 a^{-3} -7 a^2 z^8+z^8 a^{-2} -3 z^8 a^{-4} -3 z^8-7 a^3 z^7+7 a z^7+30 z^7 a^{-1} +15 z^7 a^{-3} -z^7 a^{-5} -5 a^4 z^6+13 a^2 z^6+13 z^6 a^{-2} +12 z^6 a^{-4} +19 z^6-3 a^5 z^5+12 a^3 z^5+4 a z^5-33 z^5 a^{-1} -18 z^5 a^{-3} +4 z^5 a^{-5} -a^6 z^4+5 a^4 z^4-9 a^2 z^4-16 z^4 a^{-2} -14 z^4 a^{-4} -17 z^4+3 a^5 z^3-11 a^3 z^3-11 a z^3+19 z^3 a^{-1} +12 z^3 a^{-3} -4 z^3 a^{-5} +a^6 z^2-a^4 z^2+7 z^2 a^{-2} +5 z^2 a^{-4} +4 z^2+4 a^3 z+4 a z-4 z a^{-1} -4 z a^{-3} +1-a z^{-1} - 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 12                       1 1 10                     2   -2 8                   4 1   3 6                 6 2     -4 4               7 4       3 2             7 6         -1 0           9 7           2 -2         6 9             3 -4       4 7               -3 -6     2 6                 4 -8   1 4                   -3 -10   2                     2 -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=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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