L11n217

L11n216

L11n218

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Link Presentations

[edit Notes on L11n217's Link Presentations]

Planar diagram presentation	X10,1,11,2 X8,9,1,10 X3,12,4,13 X15,22,16,9 X17,3,18,2 X21,4,22,5 X5,15,6,14 X13,21,14,20 X11,16,12,17 X19,7,20,6 X7,19,8,18
Gauss code	{1, 5, -3, 6, -7, 10, -11, -2}, {2, -1, -9, 3, -8, 7, -4, 9, -5, 11, -10, 8, -6, 4}

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)-1) (t(2)-1) \left(t(2) t(1)^2+2 t(2)^2 t(1)-3 t(2) t(1)+2 t(1)+t(2)\right)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^{5/2}-6 q^{3/2}+8 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^5+z a^5-z^5 a^3-z^3 a^3-2 z^5 a-5 z^3 a-3 z a+a z^{-1} +2 z^3 a^{-1} +2 z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^5-2 a^7 z^3+3 a^6 z^6-6 a^6 z^4+2 a^6 z^2+5 a^5 z^7-11 a^5 z^5+8 a^5 z^3-2 a^5 z+5 a^4 z^8-11 a^4 z^6+11 a^4 z^4-4 a^4 z^2+2 a^3 z^9+2 a^3 z^7-8 a^3 z^5+6 a^3 z^3+8 a^2 z^8-19 a^2 z^6+23 a^2 z^4+3 z^4 a^{-2} -11 a^2 z^2-3 z^2 a^{-2} +2 a z^9-2 a z^7+z^7 a^{-1} +9 a z^5+5 z^5 a^{-1} -15 a z^3-11 z^3 a^{-1} +6 a z+4 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +3 z^8-5 z^6+9 z^4-8 z^2-1 }[/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 6                   2 -2 4                 4   4 2               4 2   -2 0             8 4     4 -2           6 6       0 -4         6 6         0 -6       4 6           2 -8     2 6             -4 -10   1 4               3 -12   2                 -2 -14 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=0 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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