L11a286

L11a285

L11a287

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Link Presentations

[edit Notes on L11a286's Link Presentations]

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

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-t(1) t(2)^3+t(2)^3-t(1)^2 t(2)^2-t(2)^2+t(1)^2 t(2)-t(1) t(2)+t(2)+t(1)\right)}{t(1)^{3/2} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{7/2}-3 q^{5/2}+5 q^{3/2}-9 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{12}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{11}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{1}{q^{15/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^3 z^7-a z^7+a^5 z^5-4 a^3 z^5-4 a z^5+z^5 a^{-1} +3 a^5 z^3-4 a^3 z^3-4 a z^3+3 z^3 a^{-1} +a^5 z-a^3 z-a z+z a^{-1} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^3+3 a^8 z^4-a^8 z^2+5 a^7 z^5-3 a^7 z^3+7 a^6 z^6-8 a^6 z^4+9 a^5 z^7-19 a^5 z^5+9 a^5 z^3-2 a^5 z+9 a^4 z^8-25 a^4 z^6+17 a^4 z^4-4 a^4 z^2+6 a^3 z^9-15 a^3 z^7+9 a^3 z^3-2 a^3 z+2 a^2 z^{10}+3 a^2 z^8+z^8 a^{-2} -35 a^2 z^6-5 z^6 a^{-2} +44 a^2 z^4+8 z^4 a^{-2} -12 a^2 z^2-4 z^2 a^{-2} +9 a z^9+3 z^9 a^{-1} -40 a z^7-16 z^7 a^{-1} +52 a z^5+28 z^5 a^{-1} -22 a z^3-18 z^3 a^{-1} +2 a z+2 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +2 z^{10}-5 z^8-8 z^6+24 z^4-11 z^2-1 }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 8                       1 -1 6                     2   2 4                   3 1   -2 2                 6 2     4 0               4 3       -1 -2             8 6         2 -4           6 6           0 -6         5 6             -1 -8       3 6               3 -10     2 5                 -3 -12   1 3                   2 -14   2                     -2 -16 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/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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