L11a206

L11a205

L11a207

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Link Presentations

[edit Notes on L11a206's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X6718 X18,15,19,16 X16,6,17,5 X4,18,5,17 X22,11,7,12 X20,13,21,14 X14,19,15,20 X12,21,13,22
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 8, -11, 9, -10, 5, -6, 7, -5, 10, -9, 11, -8}

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{3 t(2)^2 t(1)^2-3 t(2) t(1)^2-3 t(2)^2 t(1)+5 t(2) t(1)-3 t(1)-3 t(2)+3}{t(1) t(2)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{7}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{4}{q^{5/2}}-\frac{3}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{2}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{5}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{7}{q^{11/2}}-\sqrt{q}+\frac{1}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^9-2 z a^9+z^5 a^7+3 z^3 a^7+2 z a^7+z^5 a^5+2 z^3 a^5+z^5 a^3+3 z^3 a^3+2 z a^3+a^3 z^{-1} -z^3 a-3 z a-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^{12}+4 z^4 a^{12}-3 z^2 a^{12}-2 z^7 a^{11}+8 z^5 a^{11}-7 z^3 a^{11}+z a^{11}-2 z^8 a^{10}+7 z^6 a^{10}-5 z^4 a^{10}+z^2 a^{10}-2 z^9 a^9+9 z^7 a^9-15 z^5 a^9+12 z^3 a^9-2 z a^9-z^{10} a^8+4 z^8 a^8-7 z^6 a^8+6 z^4 a^8-2 z^2 a^8-3 z^9 a^7+15 z^7 a^7-29 z^5 a^7+19 z^3 a^7-3 z a^7-z^{10} a^6+5 z^8 a^6-12 z^6 a^6+11 z^4 a^6-6 z^2 a^6-z^9 a^5+3 z^7 a^5-4 z^5 a^5-z^3 a^5+z a^5-z^8 a^4+2 z^6 a^4-2 z^4 a^4+z^2 a^4-z^7 a^3+z^5 a^3+3 z^3 a^3-3 z a^3+a^3 z^{-1} -z^6 a^2+2 z^4 a^2+z^2 a^2-a^2-z^5 a+4 z^3 a-4 z a+a 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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                         0 -2                   3 1   2 -4                 2 1     -1 -6               4 2       2 -8             4 3         -1 -10           3 3           0 -12         3 4             1 -14       2 3               -1 -16     1 3                 2 -18   1 2                   -1 -20   1                     1 -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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/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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