L10a6

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Link Presentations

[edit Notes on L10a6's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X12,6,13,5 X8493 X16,13,17,14 X14,9,15,10 X10,15,11,16 X20,12,5,11 X2,18,3,17
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -5, 7, -8, 9, -4, 6, -7, 8, -6, 10, -2, 3, -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^4-3 v^3+5 v^2-3 v+1\right)}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{8}{q^{9/2}}-q^{7/2}+\frac{13}{q^{7/2}}+4 q^{5/2}-\frac{17}{q^{5/2}}-9 q^{3/2}+\frac{17}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{4}{q^{11/2}}+13 \sqrt{q}-\frac{17}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^3+a^5 z-2 a^3 z^5-5 a^3 z^3-4 a^3 z+a z^7+4 a z^5-z^5 a^{-1} +7 a z^3-2 z^3 a^{-1} +5 a z-2 z a^{-1} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^5-a^7 z^3+4 a^6 z^6-6 a^6 z^4+3 a^6 z^2+7 a^5 z^7-11 a^5 z^5+7 a^5 z^3-2 a^5 z+6 a^4 z^8-a^4 z^6-12 a^4 z^4+8 a^4 z^2+2 a^3 z^9+16 a^3 z^7-39 a^3 z^5+z^5 a^{-3} +28 a^3 z^3-z^3 a^{-3} -8 a^3 z+13 a^2 z^8-14 a^2 z^6+4 z^6 a^{-2} -7 a^2 z^4-5 z^4 a^{-2} +7 a^2 z^2+2 z^2 a^{-2} +2 a z^9+17 a z^7+8 z^7 a^{-1} -42 a z^5-14 z^5 a^{-1} +31 a z^3+10 z^3 a^{-1} -10 a z-4 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +7 z^8-5 z^6-6 z^4+4 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 4 χ 8                     1 1 6                   3   -3 4                 6 1   5 2               7 3     -4 0             10 6       4 -2           9 9         0 -4         8 8           0 -6       5 9             4 -8     3 8               -5 -10   1 5                 4 -12   3                   -3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/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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