L10a120

L10a119

L10a121

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Link Presentations

[edit Notes on L10a120's Link Presentations]

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

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