L11n65

L11n64

L11n66

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Link Presentations

[edit Notes on L11n65's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,14,8,15 X17,20,18,21 X11,18,12,19 X19,12,20,13 X15,22,16,5 X21,16,22,17 X13,8,14,9 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -5, 6, -9, 3, -7, 8, -4, 5, -6, 4, -8, 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{2 t(1) t(2)^3+t(2)^3-2 t(1) t(2)^2-2 t(2)+t(1)+2}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{3}{q^{9/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{25/2}}-\frac{1}{q^{23/2}}+\frac{1}{q^{21/2}}-\frac{2}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{3}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{13} z^{-1} +2 a^{11} z+2 a^{11} z^{-1} -a^9 z^{-1} -a^7 z^5-3 a^7 z^3+a^7 z^{-1} -a^5 z^5-4 a^5 z^3-4 a^5 z-a^5 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{14} z^8-7 a^{14} z^6+15 a^{14} z^4-12 a^{14} z^2+3 a^{14}+a^{13} z^9-7 a^{13} z^7+14 a^{13} z^5-10 a^{13} z^3+4 a^{13} z-a^{13} z^{-1} +2 a^{12} z^8-16 a^{12} z^6+37 a^{12} z^4-29 a^{12} z^2+7 a^{12}+a^{11} z^9-8 a^{11} z^7+19 a^{11} z^5-17 a^{11} z^3+9 a^{11} z-2 a^{11} z^{-1} +a^{10} z^8-8 a^{10} z^6+20 a^{10} z^4-16 a^{10} z^2+4 a^{10}+2 a^9 z^5-4 a^9 z^3+3 a^9 z-a^9 z^{-1} +2 a^8 z^6-4 a^8 z^4+a^7 z^7-2 a^7 z^5-a^7 z^3+2 a^7 z-a^7 z^{-1} +a^6 z^6-2 a^6 z^4-a^6 z^2+a^6+a^5 z^5-4 a^5 z^3+4 a^5 z-a^5 z^{-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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     1 1 0 -8                   2     2 -10                 1 1     0 -12             1 3 2       0 -14             1 2         1 -16         1 3 2           0 -18       1   1             2 -20       1 2               -1 -22   1 1                   0 -24                         0 -26 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-11 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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