L8n6

L8n5

L8n7

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n6 at Knotilus!
	L8n6 is [math]\displaystyle{ 8^3_{10} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L8n6's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,16,12,13 X7,14,8,15 X13,8,14,9 X15,12,16,5 X2536 X4,9,1,10
Gauss code	{1, -7, 2, -8}, {-5, 4, -6, 3}, {7, -1, -4, 5, 8, -2, -3, 6}

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{(t(3)-1) (t(3)+1) (t(1) t(2)+t(3))}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-2} + q^{-6} + q^{-7} + q^{-9} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{10} z^{-2} -2 a^8 z^{-2} -2 a^8+a^6 z^{-2} +z^4 a^4+4 z^2 a^4+2 a^4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^6-6 a^{10} z^4+10 a^{10} z^2+a^{10} z^{-2} -6 a^{10}+a^9 z^5-6 a^9 z^3+8 a^9 z-2 a^9 z^{-1} +a^8 z^6-7 a^8 z^4+14 a^8 z^2+2 a^8 z^{-2} -9 a^8+a^7 z^5-6 a^7 z^3+8 a^7 z-2 a^7 z^{-1} +a^6 z^{-2} -2 a^6+a^4 z^4-4 a^4 z^2+2 a^4 }[/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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                 1 1 -5                 1 1 -7           1 1     0 -9                   0 -11       1 3 1       1 -13         2         2 -15     1             1 -17 1                 1 -19 1                 1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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