L11a479

L11a478

L11a480

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Link Presentations

[edit Notes on L11a479's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X18,7,19,8 X22,15,17,16 X20,10,21,9 X8,13,9,14 X14,17,15,18 X16,21,5,22 X12,20,13,19 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {7, -3, 9, -5, 8, -4}, {10, -1, 3, -6, 5, -2, 11, -9, 6, -7, 4, -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) (t(3)-1)^3 (t(2)+t(3))}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^4-4 q^3+9 q^2-14 q+19-20 q^{-1} +21 q^{-2} -16 q^{-3} +13 q^{-4} -7 q^{-5} +3 q^{-6} - q^{-7} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^6-a^6+2 z^4 a^4+3 z^2 a^4+a^4 z^{-2} +2 a^4-z^6 a^2-z^4 a^2-2 a^2 z^{-2} -a^2-z^6-2 z^4-3 z^2+ z^{-2} -1+z^4 a^{-2} +z^2 a^{-2} + a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^7-4 a^7 z^5+5 a^7 z^3-2 a^7 z+3 a^6 z^8-11 a^6 z^6+14 a^6 z^4-9 a^6 z^2+3 a^6+4 a^5 z^9-10 a^5 z^7+a^5 z^5+11 a^5 z^3-7 a^5 z+2 a^4 z^{10}+6 a^4 z^8-37 a^4 z^6+48 a^4 z^4+z^4 a^{-4} -27 a^4 z^2-a^4 z^{-2} +9 a^4+12 a^3 z^9-28 a^3 z^7+7 a^3 z^5+4 z^5 a^{-3} +15 a^3 z^3-z^3 a^{-3} -10 a^3 z+2 a^3 z^{-1} +2 a^2 z^{10}+16 a^2 z^8-54 a^2 z^6+9 z^6 a^{-2} +48 a^2 z^4-8 z^4 a^{-2} -20 a^2 z^2+3 z^2 a^{-2} -2 a^2 z^{-2} +7 a^2- a^{-2} +8 a z^9-4 a z^7+13 z^7 a^{-1} -19 a z^5-17 z^5 a^{-1} +17 a z^3+7 z^3 a^{-1} -6 a z-z a^{-1} +2 a z^{-1} +13 z^8-19 z^6+5 z^4+z^2- 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                       1 1 7                     3   -3 5                   6 1   5 3                 8 3     -5 1               11 6       5 -1             11 10         -1 -3           10 9           1 -5         8 13             5 -7       5 8               -3 -9     2 8                 6 -11   1 5                   -4 -13   2                     2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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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