L11n191

L11n190

L11n192

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Link Presentations

[edit Notes on L11n191's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X22,10,7,9 X10,14,11,13 X16,6,17,5 X15,21,16,20 X21,19,22,18 X19,15,20,14 X2738 X4,11,5,12 X6,18,1,17
Gauss code	{1, -9, 2, -10, 5, -11}, {9, -1, 3, -4, 10, -2, 4, 8, -6, -5, 11, 7, -8, 6, -7, -3}

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{t(1)^2 t(2)^4-2 t(1) t(2)^4+t(2)^4-2 t(1)^2 t(2)^3+2 t(1) t(2)^3-t(2)^3+t(1)^2 t(2)^2-t(1) t(2)^2+t(2)^2-t(1)^2 t(2)+2 t(1) t(2)-2 t(2)+t(1)^2-2 t(1)+1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 5 q^{9/2}-6 q^{7/2}+7 q^{5/2}-\frac{1}{q^{5/2}}-7 q^{3/2}+\frac{2}{q^{3/2}}+q^{13/2}-3 q^{11/2}+5 \sqrt{q}-\frac{5}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^{-7} -z^5 a^{-5} -4 z^3 a^{-5} -3 z a^{-5} +z^7 a^{-3} +5 z^5 a^{-3} +8 z^3 a^{-3} +6 z a^{-3} + a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-8 z^3 a^{-1} +3 a z-8 z a^{-1} +2 a z^{-1} -3 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^2 a^{-8} +3 z^3 a^{-7} -z a^{-7} +z^6 a^{-6} +z^4 a^{-6} -z^2 a^{-6} +3 z^7 a^{-5} -8 z^5 a^{-5} +8 z^3 a^{-5} -3 z a^{-5} +3 z^8 a^{-4} -8 z^6 a^{-4} +4 z^4 a^{-4} - a^{-4} +z^9 a^{-3} +3 z^7 a^{-3} -21 z^5 a^{-3} +23 z^3 a^{-3} -8 z a^{-3} + a^{-3} z^{-1} +5 z^8 a^{-2} -17 z^6 a^{-2} +10 z^4 a^{-2} +4 z^2 a^{-2} -3 a^{-2} +z^9 a^{-1} +a z^7+z^7 a^{-1} -5 a z^5-18 z^5 a^{-1} +9 a z^3+27 z^3 a^{-1} -7 a z-13 z a^{-1} +2 a z^{-1} +3 a^{-1} z^{-1} +2 z^8-8 z^6+7 z^4+2 z^2-3 }[/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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 14                   1 -1 12                 2   2 10               3 1   -2 8             3 2     1 6           4 3       -1 4         3 3         0 2       3 5           2 0     2 2             0 -2   1 4               3 -4   1                 -1 -6 1                   1

Integral Khovanov Homology (db, data source)

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