L11n61

L11n60

L11n62

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Link Presentations

[edit Notes on L11n61's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X18,11,19,12 X19,5,20,22 X15,21,16,20 X21,17,22,16 X12,17,13,18 X8,14,9,13 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 4, -8, 9, -3, -6, 7, 8, -4, -5, 6, -7, 5}

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(2)^5+t(1) t(2)^4-3 t(2)^4-2 t(1) t(2)^3+2 t(2)^3+2 t(1) t(2)^2-2 t(2)^2-3 t(1) t(2)+t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -2 q^{5/2}+4 q^{3/2}-5 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z+2 a^5 z^{-1} -3 a^3 z^3-8 a^3 z-4 a^3 z^{-1} +2 a z^5+8 a z^3-2 z^3 a^{-1} +9 a z+3 a z^{-1} -4 z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^7-5 a^5 z^5+9 a^5 z^3-7 a^5 z+2 a^5 z^{-1} +2 a^4 z^8-8 a^4 z^6+7 a^4 z^4+a^4 z^2-2 a^4+a^3 z^9+a^3 z^7-19 a^3 z^5+29 a^3 z^3-16 a^3 z+z a^{-3} +4 a^3 z^{-1} +5 a^2 z^8-19 a^2 z^6+15 a^2 z^4+z^4 a^{-2} +a^2 z^2+z^2 a^{-2} -3 a^2- a^{-2} +a z^9+2 a z^7+2 z^7 a^{-1} -21 a z^5-7 z^5 a^{-1} +28 a z^3+8 z^3 a^{-1} -13 a z-3 z a^{-1} +3 a z^{-1} + a^{-1} z^{-1} +3 z^8-11 z^6+9 z^4+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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 6                 2 2 4               2   -2 2             3 2   1 0           4 3     -1 -2         2 3 1     0 -4       3 4         1 -6     2 2           0 -8   1 4             3 -10   1               -1 -12 1                 1

Integral Khovanov Homology (db, data source)

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