L11a487

L11a486

L11a488

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Link Presentations

[edit Notes on L11a487's Link Presentations]

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

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 \left(t(3)^2-t(3)+1\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^3-5 q^2+12 q-18+28 q^{-1} -30 q^{-2} +32 q^{-3} -27 q^{-4} +20 q^{-5} -13 q^{-6} +5 q^{-7} - q^{-8} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^6 z^4-a^6 z^2-a^6 z^{-2} -a^6+2 a^4 z^6+5 a^4 z^4+4 a^4 z^2+4 a^4 z^{-2} +4 a^4-a^2 z^8-4 a^2 z^6-6 a^2 z^4-4 a^2 z^2-5 a^2 z^{-2} -5 a^2+z^6+2 z^4+z^2+2 z^{-2} +2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^5+5 a^8 z^6-2 a^8 z^4+13 a^7 z^7-16 a^7 z^5+7 a^7 z^3-3 a^7 z+a^7 z^{-1} +18 a^6 z^8-27 a^6 z^6+14 a^6 z^4-4 a^6 z^2-a^6 z^{-2} +a^6+13 a^5 z^9-4 a^5 z^7-23 a^5 z^5+18 a^5 z^3-10 a^5 z+5 a^5 z^{-1} +4 a^4 z^{10}+28 a^4 z^8-72 a^4 z^6+49 a^4 z^4-13 a^4 z^2-4 a^4 z^{-2} +4 a^4+24 a^3 z^9-38 a^3 z^7+3 a^3 z^5+14 a^3 z^3-12 a^3 z+9 a^3 z^{-1} +4 a^2 z^{10}+21 a^2 z^8-65 a^2 z^6+z^6 a^{-2} +51 a^2 z^4-z^4 a^{-2} -13 a^2 z^2-5 a^2 z^{-2} +5 a^2+11 a z^9-16 a z^7+5 z^7 a^{-1} +a z^5-8 z^5 a^{-1} +5 a z^3+2 z^3 a^{-1} -5 a z+5 a z^{-1} +11 z^8-24 z^6+17 z^4-4 z^2-2 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 7                       1 1 5                     4   -4 3                   8 1   7 1                 10 4     -6 -1               18 8       10 -3             16 14         -2 -5           16 14           2 -7         11 16             5 -9       9 16               -7 -11     4 11                 7 -13   1 9                   -8 -15   4                     4 -17 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/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}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{16} }[/math] [math]\displaystyle{ {\mathbb Z}^{16} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{16} }[/math] [math]\displaystyle{ {\mathbb Z}^{16} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{18} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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