L11a322

L11a321

L11a323

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Link Presentations

[edit Notes on L11a322's Link Presentations]

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

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(1) t(2)^4+t(1)^2 t(2)^3-2 t(1) t(2)^3+t(2)^3-2 t(1)^2 t(2)^2+3 t(1) t(2)^2-2 t(2)^2+t(1)^2 t(2)-2 t(1) t(2)+t(2)+t(1)\right)}{t(1)^{3/2} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 4 q^{9/2}-\frac{4}{q^{9/2}}-8 q^{7/2}+\frac{9}{q^{7/2}}+13 q^{5/2}-\frac{15}{q^{5/2}}-19 q^{3/2}+\frac{19}{q^{3/2}}-q^{11/2}+\frac{1}{q^{11/2}}+21 \sqrt{q}-\frac{22}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^7+z^7 a^{-1} -a^3 z^5+3 a z^5+3 z^5 a^{-1} -z^5 a^{-3} -2 a^3 z^3+3 a z^3+2 z^3 a^{-1} -2 z^3 a^{-3} -a^3 z+2 a z-z a^{-1} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -3 z^{10} a^{-2} -3 z^{10}-9 a z^9-15 z^9 a^{-1} -6 z^9 a^{-3} -14 a^2 z^8-2 z^8 a^{-2} -4 z^8 a^{-4} -12 z^8-14 a^3 z^7+6 a z^7+41 z^7 a^{-1} +20 z^7 a^{-3} -z^7 a^{-5} -9 a^4 z^6+21 a^2 z^6+28 z^6 a^{-2} +14 z^6 a^{-4} +44 z^6-4 a^5 z^5+20 a^3 z^5+17 a z^5-30 z^5 a^{-1} -20 z^5 a^{-3} +3 z^5 a^{-5} -a^6 z^4+7 a^4 z^4-7 a^2 z^4-31 z^4 a^{-2} -14 z^4 a^{-4} -32 z^4+a^5 z^3-11 a^3 z^3-14 a z^3+6 z^3 a^{-1} +6 z^3 a^{-3} -2 z^3 a^{-5} -2 a^4 z^2+8 z^2 a^{-2} +3 z^2 a^{-4} +7 z^2+2 a^3 z+4 a z+2 z a^{-1} +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                     3   -3 8                   5 1   4 6                 8 3     -5 4               11 5       6 2             10 8         -2 0           12 11           1 -2         9 12             3 -4       6 10               -4 -6     3 9                 6 -8   1 6                   -5 -10   3                     3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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