L10a119

L10a118

L10a120

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Link Presentations

[edit Notes on L10a119's Link Presentations]

Planar diagram presentation	X12,1,13,2 X20,7,11,8 X14,3,15,4 X6,15,7,16 X16,5,17,6 X4,17,5,18 X18,9,19,10 X2,11,3,12 X10,13,1,14 X8,19,9,20
Gauss code	{1, -8, 3, -6, 5, -4, 2, -10, 7, -9}, {8, -1, 9, -3, 4, -5, 6, -7, 10, -2}

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