L8a6

L8a5

L8a7

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a6 at Knotilus!
	L8a6 is [math]\displaystyle{ 8^2_{6} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a6's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X16,8,5,7 X14,10,15,9 X10,14,11,13 X8,16,9,15 X2536 X4,11,1,12
Gauss code	{1, -7, 2, -8}, {7, -1, 3, -6, 4, -5, 8, -2, 5, -4, 6, -3}

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{2 u v-3 u-3 v+2}{\sqrt{u} \sqrt{v}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{9/2}}-q^{7/2}+\frac{1}{q^{7/2}}+2 q^{5/2}-\frac{3}{q^{5/2}}-2 q^{3/2}+\frac{3}{q^{3/2}}+3 \sqrt{q}-\frac{4}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^{-1} -2 a^3 z-a^3 z^{-1} -z a^{-3} +a z^3+z^3 a^{-1} +z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a z^7-z^7 a^{-1} -a^2 z^6-2 z^6 a^{-2} -3 z^6-a^3 z^5+2 a z^5+2 z^5 a^{-1} -z^5 a^{-3} -a^4 z^4+7 z^4 a^{-2} +8 z^4-a^5 z^3-a^3 z^3-3 a z^3+3 z^3 a^{-3} -5 z^2 a^{-2} -5 z^2+2 a^5 z+2 a^3 z+a z-z a^{-3} +a^4-a^5 z^{-1} -a^3 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   \ -4 -3 -2 -1 0 1 2 3 4 χ 8                 1 1 6               1   -1 4             1 1   0 2           2 1     -1 0         2 1       1 -2       2 3         1 -4     1 1           0 -6     2             2 -8 1 1               0 -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{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/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}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/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=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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