L11n6

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Link Presentations

[edit Notes on L11n6's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X9,14,10,15 X8493 X5,11,6,10 X11,19,12,18 X13,20,14,21 X19,5,20,22 X21,12,22,13 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {-6, -1, 2, -5, -4, 6, -7, 10, -8, 4, 11, -2, 3, 7, -9, 8, -10, 9}

A Braid Representative
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{(u-1) (v-1) \left(v^2-3 v+1\right)}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-\frac{1}{q^{9/2}}-3 q^{7/2}+\frac{2}{q^{7/2}}+4 q^{5/2}-\frac{4}{q^{5/2}}-6 q^{3/2}+\frac{6}{q^{3/2}}+6 \sqrt{q}-\frac{7}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^{-1} -3 z a^3-3 a^3 z^{-1} +3 z^3 a+6 z a+4 a z^{-1} -z^5 a^{-1} -3 z^3 a^{-1} -4 z a^{-1} -2 a^{-1} z^{-1} +z^3 a^{-3} +z a^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^3-2 a^5 z+a^5 z^{-1} +z^6 a^{-4} +2 a^4 z^4-3 z^4 a^{-4} -3 a^4 z^2+z^2 a^{-4} +a^4+a^3 z^7+3 z^7 a^{-3} -4 a^3 z^5-11 z^5 a^{-3} +12 a^3 z^3+9 z^3 a^{-3} -12 a^3 z-3 z a^{-3} +3 a^3 z^{-1} +2 a^2 z^8+3 z^8 a^{-2} -9 a^2 z^6-10 z^6 a^{-2} +18 a^2 z^4+6 z^4 a^{-2} -12 a^2 z^2-z^2 a^{-2} +3 a^2+ a^{-2} +a z^9+z^9 a^{-1} +2 z^7 a^{-1} -13 a z^5-20 z^5 a^{-1} +30 a z^3+28 z^3 a^{-1} -20 a z-13 z a^{-1} +4 a z^{-1} +2 a^{-1} z^{-1} +5 z^8-20 z^6+25 z^4-11 z^2+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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 2   2 6               2 1   -1 4             4 2     2 2         1 3 2       0 0         5 4         1 -2       3 4           1 -4     1 3             -2 -6   1 3               2 -8   1                 -1 -10 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{ i=2 }[/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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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