L10a58

L10a57

L10a59

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Link Presentations

[edit Notes on L10a58's Link Presentations]

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

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