L11n60

L11n59

L11n61

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Link Presentations

[edit Notes on L11n60's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X11,19,12,18 X22,19,5,20 X20,15,21,16 X16,21,17,22 X17,13,18,12 X8,14,9,13 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, -4, 8, 9, -3, 6, -7, -8, 4, 5, -6, 7, -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{t(2)^5+t(1) t(2)^4-3 t(2)^4-4 t(1) t(2)^3+6 t(2)^3+6 t(1) t(2)^2-4 t(2)^2-3 t(1) t(2)+t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{3/2}+4 \sqrt{q}-\frac{7}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{6}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^7-a^7 z^{-1} +3 z^3 a^5+6 z a^5+4 a^5 z^{-1} -2 z^5 a^3-7 z^3 a^3-10 z a^3-4 a^3 z^{-1} +3 z^3 a+4 z a+a z^{-1} -z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^6-3 a^8 z^4+3 a^8 z^2-a^8+3 a^7 z^7-9 a^7 z^5+8 a^7 z^3-4 a^7 z+a^7 z^{-1} +3 a^6 z^8-4 a^6 z^6-9 a^6 z^4+11 a^6 z^2-4 a^6+a^5 z^9+7 a^5 z^7-29 a^5 z^5+29 a^5 z^3-15 a^5 z+4 a^5 z^{-1} +6 a^4 z^8-11 a^4 z^6-4 a^4 z^4+14 a^4 z^2-7 a^4+a^3 z^9+6 a^3 z^7-23 a^3 z^5+28 a^3 z^3-14 a^3 z+4 a^3 z^{-1} +3 a^2 z^8-6 a^2 z^6+6 a^2 z^4+4 a^2 z^2-4 a^2+2 a z^7-3 a z^5+8 a z^3+z^3 a^{-1} -4 a z-z a^{-1} +a z^{-1} +4 z^4-2 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 4                   1 1 2                 3   -3 0               4 1   3 -2             6 4     -2 -4           5 3       2 -6         4 6         2 -8       5 5           0 -10     2 5             3 -12   1 4               -3 -14   2                 2 -16 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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/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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