L11n193

L11n192

L11n194

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Link Presentations

[edit Notes on L11n193's Link Presentations]

Planar diagram presentation	X8192 X9,21,10,20 X4758 X16,5,17,6 X6,15,1,16 X22,17,7,18 X18,13,19,14 X14,21,15,22 X2,11,3,12 X12,3,13,4 X19,11,20,10
Gauss code	{1, -9, 10, -3, 4, -5}, {3, -1, -2, 11, 9, -10, 7, -8, 5, -4, 6, -7, -11, 2, 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) t(2)^4-t(2)^4+t(1)^2 t(2)^3-5 t(1) t(2)^3+3 t(2)^3-4 t(1)^2 t(2)^2+7 t(1) t(2)^2-4 t(2)^2+3 t(1)^2 t(2)-5 t(1) t(2)+t(2)-t(1)^2+t(1)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{2}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{9}{q^{7/2}}+\frac{11}{q^{9/2}}-\frac{13}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{10}{q^{15/2}}+\frac{7}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{1}{q^{21/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^9+z^5 a^7+z^3 a^7+z a^7+a^7 z^{-1} +z^5 a^5-2 z a^5-a^5 z^{-1} -2 z^3 a^3-2 z a^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{12} z^6-2 a^{12} z^4+a^{12} z^2+4 a^{11} z^7-11 a^{11} z^5+7 a^{11} z^3+5 a^{10} z^8-12 a^{10} z^6+4 a^{10} z^4+2 a^{10} z^2+2 a^9 z^9+4 a^9 z^7-20 a^9 z^5+12 a^9 z^3-2 a^9 z+9 a^8 z^8-19 a^8 z^6+8 a^8 z^4-a^8 z^2+2 a^7 z^9+3 a^7 z^7-9 a^7 z^5+a^7 z^3+3 a^7 z-a^7 z^{-1} +4 a^6 z^8-5 a^6 z^6+6 a^6 z^4-4 a^6 z^2+a^6+3 a^5 z^7-a^5 z^3+3 a^5 z-a^5 z^{-1} +a^4 z^6+4 a^4 z^4-2 a^4 z^2+3 a^3 z^3-2 a^3 z }[/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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                   2 2 -4                 4 1 -3 -6               5 1   4 -8             6 4     -2 -10           7 5       2 -12         5 6         1 -14       5 7           -2 -16     3 6             3 -18   1 4               -3 -20   3                 3 -22 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=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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