L11a236

L11a235

L11a237

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Link Presentations

[edit Notes on L11a236's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X20,15,21,16 X16,9,17,10 X10,19,11,20 X22,13,7,14 X14,21,15,22 X18,5,19,6 X2738 X4,11,5,12 X6,17,1,18
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 4, -5, 10, -2, 6, -7, 3, -4, 11, -8, 5, -3, 7, -6}

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{5 u^2 v^2-6 u^2 v+2 u^2+5 u v^3-9 u v^2+5 u v+2 v^4-6 v^3+5 v^2}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{9}{q^{9/2}}-\frac{13}{q^{11/2}}+\frac{13}{q^{13/2}}-\frac{14}{q^{15/2}}+\frac{12}{q^{17/2}}-\frac{9}{q^{19/2}}+\frac{6}{q^{21/2}}-\frac{3}{q^{23/2}}+\frac{1}{q^{25/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{13} z^{-1} +4 a^{11} z+2 a^{11} z^{-1} -3 a^9 z^3-a^9 z-5 a^7 z^3-5 a^7 z-a^7 z^{-1} -3 a^5 z^3-a^5 z-a^3 z^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{14} z^8-5 a^{14} z^6+9 a^{14} z^4-7 a^{14} z^2+2 a^{14}+3 a^{13} z^9-15 a^{13} z^7+24 a^{13} z^5-13 a^{13} z^3+2 a^{13} z-a^{13} z^{-1} +2 a^{12} z^{10}-3 a^{12} z^8-18 a^{12} z^6+44 a^{12} z^4-29 a^{12} z^2+5 a^{12}+10 a^{11} z^9-42 a^{11} z^7+51 a^{11} z^5-21 a^{11} z^3+7 a^{11} z-2 a^{11} z^{-1} +2 a^{10} z^{10}+7 a^{10} z^8-50 a^{10} z^6+64 a^{10} z^4-25 a^{10} z^2+3 a^{10}+7 a^9 z^9-15 a^9 z^7-6 a^9 z^5+14 a^9 z^3-2 a^9 z+11 a^8 z^8-28 a^8 z^6+15 a^8 z^4-2 a^8 z^2-a^8+12 a^7 z^7-27 a^7 z^5+17 a^7 z^3-6 a^7 z+a^7 z^{-1} +9 a^6 z^6-11 a^6 z^4+a^6 z^2+6 a^5 z^5-4 a^5 z^3+a^5 z+3 a^4 z^4+a^3 z^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 6         0 -14           8 7           1 -16         5 7             2 -18       4 7               -3 -20     2 5                 3 -22   1 4                   -3 -24   2                     2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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}^{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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