L10a126

L10a125

L10a127

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Link Presentations

[edit Notes on L10a126's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,7,17,8 X8,15,5,16 X20,12,9,11 X18,14,19,13 X14,18,15,17 X12,20,13,19 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -4}, {10, -2, 5, -8, 6, -7, 4, -3, 7, -6, 8, -5}

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{-3 t(2) t(1)+2 t(2) t(3) t(1)-3 t(3) t(1)+3 t(1)+3 t(2)-3 t(2) t(3)+3 t(3)-2}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-7} - q^{-6} +4 q^{-5} -4 q^{-4} +q^3+7 q^{-3} -3 q^2-7 q^{-2} +4 q+7 q^{-1} -5 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^8 z^{-2} -2 a^6 z^{-2} -3 a^6+3 z^2 a^4+a^4 z^{-2} +3 a^4-z^4 a^2-z^4-z^2+z^2 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^3 z^9+a z^9+a^4 z^8+4 a^2 z^8+3 z^8+a^5 z^7-2 a^3 z^7+3 z^7 a^{-1} +a^6 z^6-13 a^2 z^6+z^6 a^{-2} -11 z^6+a^7 z^5+a^5 z^5+4 a^3 z^5-7 a z^5-11 z^5 a^{-1} +a^8 z^4+2 a^6 z^4+a^4 z^4+13 a^2 z^4-3 z^4 a^{-2} +10 z^4-3 a^3 z^3+5 a z^3+8 z^3 a^{-1} -3 a^8 z^2-6 a^6 z^2-3 a^4 z^2-3 a^2 z^2+z^2 a^{-2} -2 z^2-3 a^7 z-3 a^5 z+3 a^8+5 a^6+3 a^4+2 a^7 z^{-1} +2 a^5 z^{-1} -a^8 z^{-2} -2 a^6 z^{-2} -a^4 z^{-2} }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                     1 1 5                   2   -2 3                 2 1   1 1               3 2     -1 -1             4 2       2 -3           4 4         0 -5         3 3           0 -7       1 4             3 -9     3 3               0 -11   1 4                 3 -13                       0 -15 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/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=4 }[/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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