L11a464

L11a463

L11a465

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Link Presentations

[edit Notes on L11a464's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X16,6,17,5 X8493 X20,9,21,10 X10,19,11,20 X22,18,15,17 X18,22,19,21 X14,16,5,15 X2,12,3,11
Gauss code	{1, -11, 5, -3}, {10, -4, 8, -9, 7, -6, 9, -8}, {4, -1, 2, -5, 6, -7, 11, -2, 3, -10}

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{2 (u-1) (v-1) (w-1)^3}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^6-5 q^5+10 q^4-14 q^3+19 q^2-20 q+21-16 q^{-1} +12 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^6 a^{-2} -z^6+2 a^2 z^4-z^4 a^{-2} +z^4 a^{-4} -2 z^4-a^4 z^2+4 a^2 z^2+2 z^2 a^{-2} -5 z^2-a^4+4 a^2+4 a^{-2} - a^{-4} -6+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-6} -z^4 a^{-6} +5 z^7 a^{-5} +a^5 z^5-10 z^5 a^{-5} -2 a^5 z^3+2 z^3 a^{-5} +a^5 z+z a^{-5} +9 z^8 a^{-4} +3 a^4 z^6-22 z^6 a^{-4} -6 a^4 z^4+11 z^4 a^{-4} +5 a^4 z^2+z^2 a^{-4} -2 a^4-2 a^{-4} +7 z^9 a^{-3} +4 a^3 z^7-11 z^7 a^{-3} -3 a^3 z^5-2 z^5 a^{-3} -3 a^3 z^3+z^3 a^{-3} +3 a^3 z+3 z a^{-3} +2 z^{10} a^{-2} +4 a^2 z^8+11 z^8 a^{-2} +3 a^2 z^6-33 z^6 a^{-2} -17 a^2 z^4+16 z^4 a^{-2} +18 a^2 z^2+10 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -8 a^2-8 a^{-2} +4 a z^9+11 z^9 a^{-1} -20 z^7 a^{-1} -3 a z^5+9 z^5 a^{-1} -a z^3-z^3 a^{-1} +4 a z+4 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +2 z^{10}+6 z^8-10 z^6-7 z^4+22 z^2+2 z^{-2} -11 }[/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 13                       1 1 11                     4   -4 9                   6 1   5 7                 8 4     -4 5               11 6       5 3             9 8         -1 1           12 11           1 -1         8 13             5 -3       4 8               -4 -5     2 8                 6 -7   1 4                   -3 -9   2                     2 -11 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=-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}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/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}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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