L11n90

L11n89

L11n91

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Link Presentations

[edit Notes on L11n90's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X7,16,8,17 X17,22,18,5 X14,9,15,10 X10,20,11,19 X21,9,22,8 X18,14,19,13 X20,15,21,16 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 7, 5, -6, 11, -2, 8, -5, 9, 3, -4, -8, 6, -9, -7, 4}

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)-2) (t(1)+t(2)) (2 t(2)-1)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{3}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{5}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z^{-1} -3 z a^7-2 a^7 z^{-1} +2 z^3 a^5+2 z a^5+a^5 z^{-1} +2 z^3 a^3+2 z a^3+a^3 z^{-1} -3 z a-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^7 a^9+5 z^5 a^9-8 z^3 a^9+5 z a^9-a^9 z^{-1} -2 z^8 a^8+9 z^6 a^8-11 z^4 a^8+4 z^2 a^8-a^8-z^9 a^7+15 z^5 a^7-25 z^3 a^7+13 z a^7-2 a^7 z^{-1} -5 z^8 a^6+21 z^6 a^6-25 z^4 a^6+13 z^2 a^6-3 a^6-z^9 a^5-z^7 a^5+15 z^5 a^5-19 z^3 a^5+9 z a^5-a^5 z^{-1} -3 z^8 a^4+12 z^6 a^4-17 z^4 a^4+12 z^2 a^4-2 a^4-2 z^7 a^3+5 z^5 a^3-2 z^3 a^3-3 z a^3+a^3 z^{-1} -3 z^4 a^2+3 z^2 a^2-a^2-4 z a+a 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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ 0                 3 3 -2               2 1 -1 -4             3 2   1 -6           3 2     -1 -8         3 3       0 -10       3 4         1 -12     1 2           -1 -14   1 3             2 -16   1               -1 -18 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{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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