L11n144

L11n143

L11n145

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Link Presentations

[edit Notes on L11n144's Link Presentations]

Planar diagram presentation	X8192 X11,19,12,18 X3,10,4,11 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X20,16,21,15 X22,14,7,13 X14,22,15,21 X19,4,20,5
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{2 t(1) t(2)^4-3 t(1) t(2)^3+2 t(2)^3-t(1)^2 t(2)^2+3 t(1) t(2)^2-t(2)^2+2 t(1)^2 t(2)-3 t(1) t(2)+2 t(1)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{4}{q^{9/2}}+\frac{5}{q^{7/2}}+q^{5/2}-\frac{7}{q^{5/2}}-2 q^{3/2}+\frac{6}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}+4 \sqrt{q}-\frac{6}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^3+2 a^5 z+a^5 z^{-1} -a^3 z^5-3 a^3 z^3-3 a^3 z-a z^5-3 a z^3+z^3 a^{-1} -4 a z-2 a z^{-1} +2 z a^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^3 z^9-a z^9-2 a^4 z^8-3 a^2 z^8-z^8-3 a^5 z^7+2 a^3 z^7+5 a z^7-2 a^6 z^6+5 a^4 z^6+12 a^2 z^6+5 z^6-a^7 z^5+10 a^5 z^5-a^3 z^5-14 a z^5-2 z^5 a^{-1} +5 a^6 z^4-3 a^4 z^4-22 a^2 z^4-z^4 a^{-2} -15 z^4+3 a^7 z^3-11 a^5 z^3-4 a^3 z^3+14 a z^3+4 z^3 a^{-1} -a^6 z^2-2 a^4 z^2+13 a^2 z^2+3 z^2 a^{-2} +17 z^2-a^7 z+6 a^5 z+2 a^3 z-7 a z-2 z a^{-1} +a^4-3 a^2-2 a^{-2} -5-a^5 z^{-1} +2 a z^{-1} + a^{-1} 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                   1 -1 4                 1   1 2               3 1   -2 0             3 1     2 -2           4 4       0 -4         3 2         1 -6       2 4           2 -8     2 3             -1 -10     2               2 -12 1 2                 -1 -14 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\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 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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