L10a77

L10a76

L10a78

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Link Presentations

[edit Notes on L10a77's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X16,5,17,6 X14,7,15,8 X18,11,19,12 X20,15,7,16 X12,17,13,18 X4,13,5,14 X6,19,1,20
Gauss code	{1, -2, 3, -9, 4, -10}, {5, -1, 2, -3, 6, -8, 9, -5, 7, -4, 8, -6, 10, -7}

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