L11n145

L11n144

L11n146

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Link Presentations

[edit Notes on L11n145's Link Presentations]

Planar diagram presentation	X8192 X18,11,19,12 X10,4,11,3 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X15,20,16,21 X13,22,14,7 X21,14,22,15 X4,20,5,19
Gauss code	{1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, 2, -5, -9, 10, -8, -7, 4, -2, 11, 8, -10, 9}

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)^2 t(2)^4-t(1) t(2)^4-3 t(1)^2 t(2)^3+4 t(1) t(2)^3-t(2)^3+3 t(1)^2 t(2)^2-5 t(1) t(2)^2+3 t(2)^2-t(1)^2 t(2)+4 t(1) t(2)-3 t(2)-t(1)+1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{3/2}-4 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{9}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{11}{q^{7/2}}+\frac{9}{q^{9/2}}-\frac{7}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{1}{q^{15/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^3 z^7+a^5 z^5-4 a^3 z^5+a z^5+2 a^5 z^3-4 a^3 z^3+2 a z^3-a z+a^3 z^{-1} -a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^3 a^9-4 z^4 a^8+2 z^2 a^8-z^7 a^7-3 z^5 a^7+2 z^3 a^7-3 z^8 a^6+7 z^6 a^6-12 z^4 a^6+6 z^2 a^6-2 z^9 a^5+2 z^7 a^5-8 z^8 a^4+23 z^6 a^4-20 z^4 a^4+6 z^2 a^4-2 z^9 a^3-z^7 a^3+15 z^5 a^3-10 z^3 a^3-z a^3+a^3 z^{-1} -5 z^8 a^2+15 z^6 a^2-10 z^4 a^2+2 z^2 a^2-a^2-4 z^7 a+12 z^5 a-7 z^3 a-z a+a z^{-1} -z^6+2 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                   1 -1 2                 3   3 0               3 1   -2 -2             6 3     3 -4           5 4       -1 -6         6 5         1 -8       4 6           2 -10     3 5             -2 -12   1 4               3 -14   3                 -3 -16 1                   1

Integral Khovanov Homology (db, data source)

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