L11a434

L11a433

L11a435

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	Brunnian link

Link Presentations

[edit Notes on L11a434's Link Presentations]

Planar diagram presentation	X6172 X16,12,17,11 X8493 X2,18,3,17 X14,6,15,5 X18,7,19,8 X12,16,5,15 X20,14,21,13 X22,9,13,10 X10,21,11,22 X4,19,1,20
Gauss code	{1, -4, 3, -11}, {5, -1, 6, -3, 9, -10, 2, -7}, {8, -5, 7, -2, 4, -6, 11, -8, 10, -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{(u-1) (v-1) (w-1) (v+w-1) (v w-v-w)}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^6-4 q^5+8 q^4-14 q^3+21 q^2-22 q+24-20 q^{-1} +16 q^{-2} -9 q^{-3} +4 q^{-4} - q^{-5} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^6 a^{-2} -z^6+2 a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} -a^4 z^2+a^2 z^2-3 z^2 a^{-2} +z^2 a^{-4} +2 z^2+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10}+7 a z^9+13 z^9 a^{-1} +6 z^9 a^{-3} +10 a^2 z^8+14 z^8 a^{-2} +7 z^8 a^{-4} +17 z^8+8 a^3 z^7-a z^7-20 z^7 a^{-1} -7 z^7 a^{-3} +4 z^7 a^{-5} +4 a^4 z^6-14 a^2 z^6-46 z^6 a^{-2} -18 z^6 a^{-4} +z^6 a^{-6} -45 z^6+a^5 z^5-11 a^3 z^5-12 a z^5+3 z^5 a^{-1} -7 z^5 a^{-3} -10 z^5 a^{-5} -5 a^4 z^4+8 a^2 z^4+48 z^4 a^{-2} +16 z^4 a^{-4} -2 z^4 a^{-6} +43 z^4-a^5 z^3+5 a^3 z^3+7 a z^3+5 z^3 a^{-1} +10 z^3 a^{-3} +6 z^3 a^{-5} +2 a^4 z^2-4 a^2 z^2-20 z^2 a^{-2} -6 z^2 a^{-4} -20 z^2+1-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2} }[/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                     3   -3 9                   5 1   4 7                 9 3     -6 5               12 5       7 3             10 9         -1 1           14 12           2 -1         10 14             4 -3       6 10               -4 -5     3 10                 7 -7   1 6                   -5 -9   3                     3 -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}^{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}^{10}\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^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/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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