L11n168

L11n167

L11n169

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Link Presentations

[edit Notes on L11n168's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X18,11,19,12 X19,7,20,22 X15,21,16,20 X21,17,22,16 X12,17,13,18 X6718 X4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 5, -9, 11, -4, -7, 8, 9, -5, -6, 7, -8, 6}

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-t(1)^2 t(2)^3+3 t(1) t(2)^3-t(2)^3+2 t(1)^2 t(2)^2-3 t(1) t(2)^2+2 t(2)^2-t(1)^2 t(2)+3 t(1) t(2)-t(2)-t(1)+1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{7}{q^{9/2}}-\frac{7}{q^{7/2}}+\frac{4}{q^{5/2}}-\frac{3}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{2}{q^{17/2}}-\frac{4}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{7}{q^{11/2}}+\frac{1}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^7+4 z^3 a^7+5 z a^7+a^7 z^{-1} -z^7 a^5-6 z^5 a^5-13 z^3 a^5-10 z a^5-a^5 z^{-1} +z^5 a^3+3 z^3 a^3+2 z a^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^5-3 a^{11} z^3+2 a^{11} z+2 a^{10} z^6-5 a^{10} z^4+2 a^{10} z^2+2 a^9 z^7-3 a^9 z^5-a^9 z^3+2 a^8 z^8-5 a^8 z^6+7 a^8 z^4-5 a^8 z^2+a^7 z^9-2 a^7 z^7+5 a^7 z^5-6 a^7 z^3+5 a^7 z-a^7 z^{-1} +3 a^6 z^8-10 a^6 z^6+18 a^6 z^4-10 a^6 z^2+a^6+a^5 z^9-4 a^5 z^7+12 a^5 z^5-14 a^5 z^3+9 a^5 z-a^5 z^{-1} +a^4 z^8-3 a^4 z^6+7 a^4 z^4-5 a^4 z^2+3 a^3 z^5-6 a^3 z^3+2 a^3 z+a^2 z^4-2 a^2 z^2 }[/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 1 χ 0                   1 -1 -2                 2   2 -4               3 2   -1 -6             4 1     3 -8           3 3       0 -10         4 4         0 -12       2 3           1 -14     2 4             -2 -16   1 3               2 -18   1                 -1 -20 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=-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}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/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

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