L11a516

L11a515

L11a517

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Link Presentations

[edit Notes on L11a516's Link Presentations]

Planar diagram presentation	X8192 X16,5,17,6 X18,9,19,10 X10,17,11,18 X20,15,21,16 X14,4,15,3 X4,22,5,21 X2738 X22,12,13,11 X12,14,7,13 X6,19,1,20
Gauss code	{1, -8, 6, -7, 2, -11}, {8, -1, 3, -4, 9, -10}, {10, -6, 5, -2, 4, -3, 11, -5, 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(3)-1) \left(t(3)^2 t(2)^2+t(1) t(3) t(2)^2-2 t(3) t(2)^2+t(2)^2+t(1) t(3)^2 t(2)-t(3)^2 t(2)+t(1) t(2)-2 t(1) t(3) t(2)+2 t(3) t(2)-t(2)-t(1) t(3)^2-t(1)+2 t(1) t(3)-t(3)\right)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^4-4 q^3+9 q^2-15 q+21-22 q^{-1} +24 q^{-2} -19 q^{-3} +15 q^{-4} -9 q^{-5} +4 q^{-6} - q^{-7} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^6-a^6+3 z^4 a^4+6 z^2 a^4+a^4 z^{-2} +3 a^4-2 z^6 a^2-6 z^4 a^2-7 z^2 a^2-2 a^2 z^{-2} -4 a^2-z^6-z^4+z^2+ z^{-2} +2+z^4 a^{-2} +z^2 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^7-3 a^7 z^5+3 a^7 z^3-a^7 z+4 a^6 z^8-13 a^6 z^6+14 a^6 z^4-7 a^6 z^2+2 a^6+6 a^5 z^9-16 a^5 z^7+7 a^5 z^5+6 a^5 z^3-3 a^5 z+3 a^4 z^{10}+7 a^4 z^8-48 a^4 z^6+60 a^4 z^4+z^4 a^{-4} -29 a^4 z^2-a^4 z^{-2} +7 a^4+17 a^3 z^9-44 a^3 z^7+24 a^3 z^5+4 z^5 a^{-3} +3 a^3 z^3-z^3 a^{-3} -4 a^3 z+2 a^3 z^{-1} +3 a^2 z^{10}+19 a^2 z^8-73 a^2 z^6+9 z^6 a^{-2} +74 a^2 z^4-7 z^4 a^{-2} -33 a^2 z^2+2 z^2 a^{-2} -2 a^2 z^{-2} +7 a^2+11 a z^9-13 a z^7+14 z^7 a^{-1} -8 a z^5-18 z^5 a^{-1} +8 a z^3+7 z^3 a^{-1} -2 a z+2 a z^{-1} +16 z^8-29 z^6+20 z^4-9 z^2- z^{-2} +3 }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                       1 1 7                     3   -3 5                   6 1   5 3                 9 3     -6 1               12 6       6 -1             12 11         -1 -3           12 10           2 -5         9 14             5 -7       6 10               -4 -9     3 9                 6 -11   1 6                   -5 -13   3                     3 -15 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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/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}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/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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