L11a120

L11a119

L11a121

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Link Presentations

[edit Notes on L11a120's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X22,15,5,16 X18,7,19,8 X10,21,11,22 X20,11,21,12 X12,19,13,20 X16,9,17,10 X8,17,9,18 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, 8, -5, 6, -7, 11, -2, 3, -8, 9, -4, 7, -6, 5, -3}

A Braid Representative	{{{braid_table}}}
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{7 u v^2-10 u v+4 u+4 v^3-10 v^2+7 v}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{9}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{3/2}}+\frac{1}{q^{25/2}}-\frac{2}{q^{23/2}}+\frac{5}{q^{21/2}}-\frac{8}{q^{19/2}}+\frac{11}{q^{17/2}}-\frac{13}{q^{15/2}}+\frac{12}{q^{13/2}}-\frac{13}{q^{11/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{13} z^{-1} +3 z a^{11}+a^{11} z^{-1} -2 z^3 a^9+2 z a^9+2 a^9 z^{-1} -5 z^3 a^7-6 z a^7-2 a^7 z^{-1} -3 z^3 a^5-z a^5-z^3 a^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^8 a^{14}+6 z^6 a^{14}-13 z^4 a^{14}+12 z^2 a^{14}-4 a^{14}-2 z^9 a^{13}+10 z^7 a^{13}-16 z^5 a^{13}+8 z^3 a^{13}+a^{13} z^{-1} -z^{10} a^{12}-2 z^8 a^{12}+27 z^6 a^{12}-54 z^4 a^{12}+38 z^2 a^{12}-9 a^{12}-7 z^9 a^{11}+26 z^7 a^{11}-22 z^5 a^{11}-2 z^3 a^{11}+2 z a^{11}+a^{11} z^{-1} -z^{10} a^{10}-11 z^8 a^{10}+54 z^6 a^{10}-65 z^4 a^{10}+26 z^2 a^{10}-4 a^{10}-5 z^9 a^9+4 z^7 a^9+28 z^5 a^9-35 z^3 a^9+11 z a^9-2 a^9 z^{-1} -10 z^8 a^8+24 z^6 a^8-10 z^4 a^8-z^2 a^8+2 a^8-12 z^7 a^7+28 z^5 a^7-20 z^3 a^7+8 z a^7-2 a^7 z^{-1} -9 z^6 a^6+11 z^4 a^6-z^2 a^6-6 z^5 a^5+4 z^3 a^5-z a^5-3 z^4 a^4-z^3 a^3 }[/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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                       1 1 -4                     3 1 -2 -6                   3     3 -8                 6 3     -3 -10               7 3       4 -12             6 7         1 -14           7 6           1 -16         4 6             2 -18       4 7               -3 -20     1 4                 3 -22   1 4                   -3 -24   1                     1 -26 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=-11 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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