L8n3

L8n2

L8n4

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n3 at Knotilus!
	L8n3 is [math]\displaystyle{ 8^3_{7} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L8n3's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X9,16,10,11 X11,10,12,5 X4,15,1,16
Gauss code	{1, 4, -3, -8}, {-2, -1, 5, 3, -6, 7}, {-7, 2, -4, -5, 8, 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{u v^2 w^2-1}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-3} + q^{-5} + q^{-7} + q^{-9} }[/math] (db)
Signature	-6 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{10} z^{-2} +a^{10}-z^4 a^8-5 z^2 a^8-2 a^8 z^{-2} -6 a^8+z^6 a^6+6 z^4 a^6+10 z^2 a^6+a^6 z^{-2} +5 a^6 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{12}+a^{10} z^2+a^{10} z^{-2} -3 a^{10}+a^9 z^5-5 a^9 z^3+6 a^9 z-2 a^9 z^{-1} +a^8 z^6-6 a^8 z^4+11 a^8 z^2+2 a^8 z^{-2} -8 a^8+a^7 z^5-5 a^7 z^3+6 a^7 z-2 a^7 z^{-1} +a^6 z^6-6 a^6 z^4+10 a^6 z^2+a^6 z^{-2} -5 a^6 }[/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 χ -5             1 1 -7             1 1 -9         1     1 -11     1         1 -13     2 1       1 -15 1             1 -17 2 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=-7 }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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 }[/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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