L11n136

L11n135

L11n137

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Link Presentations

[edit Notes on L11n136's Link Presentations]

Planar diagram presentation	X8192 X11,19,12,18 X3,10,4,11 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X20,14,21,13 X22,16,7,15 X19,4,20,5 X14,22,15,21
Gauss code	{1, -4, -3, 10, 5, -6}, {6, -1, 7, 3, -2, -5, 8, -11, 9, -7, 4, 2, -10, -8, 11, -9}

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) t(2)^6-t(1) t(2)^5+t(2)^5+t(1) t(2)^4-t(1) t(2)^3+t(1) t(2)^2+t(1)^2 t(2)-t(1) t(2)+t(1)}{t(1) t(2)^3} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{1}{q^{9/2}}+q^{7/2}-\frac{2}{q^{7/2}}-q^{5/2}+\frac{2}{q^{5/2}}+2 q^{3/2}-\frac{3}{q^{3/2}}-\frac{1}{q^{11/2}}-3 \sqrt{q}+\frac{2}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^5+5 a^3 z^3+7 a^3 z+3 a^3 z^{-1} -a z^7-7 a z^5+z^5 a^{-1} -17 a z^3+5 z^3 a^{-1} -17 a z+7 z a^{-1} -5 a z^{-1} +2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^3 z^9-a z^9-a^4 z^8-2 a^2 z^8-z^8-a^5 z^7+6 a^3 z^7+7 a z^7+5 a^4 z^6+11 a^2 z^6+6 z^6+6 a^5 z^5-13 a^3 z^5-21 a z^5-2 z^5 a^{-1} -6 a^4 z^4-20 a^2 z^4-z^4 a^{-2} -15 z^4-10 a^5 z^3+16 a^3 z^3+33 a z^3+6 z^3 a^{-1} -z^3 a^{-3} +a^4 z^2+17 a^2 z^2-z^2 a^{-4} +17 z^2+4 a^5 z-11 a^3 z-22 a z-7 z a^{-1} -5 a^2+ a^{-4} -5+3 a^3 z^{-1} +5 a z^{-1} +2 a^{-1} z^{-1} }[/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 χ 8                   1 -1 6                     0 4               2 1   -1 2             1       1 0           2 3       1 -2         1           1 -4       1 2           1 -6     1 1             0 -8     1               1 -10 1 1                 0 -12 1                   1

Integral Khovanov Homology (db, data source)

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