L11n186

L11n185

L11n187

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Link Presentations

[edit Notes on L11n186's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X18,10,19,9 X19,22,20,7 X13,20,14,21 X21,14,22,15 X10,16,11,15 X16,6,17,5 X2738 X4,11,5,12 X6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 3, -7, 10, -2, -5, 6, 7, -8, 11, -3, -4, 5, -6, 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{\left(t(1) t(2)^2-t(1) t(2)+t(2)+t(1)-1\right) \left(t(1) t(2)^2-t(2)^2-t(1) t(2)+t(2)-1\right)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{7/2}+3 q^{5/2}-5 q^{3/2}+7 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{8}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{1}{q^{11/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^7-a^3 z^5+5 a z^5-z^5 a^{-1} -3 a^3 z^3+8 a z^3-3 z^3 a^{-1} -2 a^3 z+3 a z-2 z a^{-1} +a^3 z^{-1} -a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^4-2 a^6 z^2+3 a^5 z^5-5 a^5 z^3+a^5 z+a^4 z^8-2 a^4 z^6+5 a^4 z^4-4 a^4 z^2+a^3 z^9-2 a^3 z^7+4 a^3 z^5+z^5 a^{-3} -2 z^3 a^{-3} -a^3 z^{-1} +4 a^2 z^8-11 a^2 z^6+3 z^6 a^{-2} +14 a^2 z^4-7 z^4 a^{-2} -4 a^2 z^2+2 z^2 a^{-2} +a^2+a z^9+2 a z^7+4 z^7 a^{-1} -10 a z^5-10 z^5 a^{-1} +14 a z^3+7 z^3 a^{-1} -4 a z-3 z a^{-1} -a z^{-1} +3 z^8-6 z^6+3 z^4 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                   1 1 6                 2   -2 4               3 1   2 2             4 2     -2 0           5 3       2 -2         4 5         1 -4       4 4           0 -6     2 5             3 -8   1 3               -2 -10   2                 2 -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{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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