L8n4

L8n3

L8n5

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n4 at Knotilus!
	L8n4 is [math]\displaystyle{ 8^3_{8} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L8n4's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X9,16,10,11 X11,10,12,5 X15,1,16,4
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{t(3) t(2)^2-t(1) t(3)^2 t(2)+t(1) t(3) t(2)-t(3) t(2)+t(2)-t(1) t(3)}{\sqrt{t(1)} t(2) t(3)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-7} - q^{-6} +2 q^{-5} -2 q^{-4} +3 q^{-3} - q^{-2} +2 q^{-1} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 z^2+a^6 z^{-2} +2 a^6-a^4 z^4-4 a^4 z^2-2 a^4 z^{-2} -6 a^4+2 a^2 z^2+a^2 z^{-2} +4 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^8-3 z^2 a^8+a^8+z^5 a^7-2 z^3 a^7+z^6 a^6-3 z^4 a^6+4 z^2 a^6+a^6 z^{-2} -3 a^6+2 z^5 a^5-5 z^3 a^5+6 z a^5-2 a^5 z^{-1} +z^6 a^4-4 z^4 a^4+10 z^2 a^4+2 a^4 z^{-2} -8 a^4+z^5 a^3-3 z^3 a^3+6 z a^3-2 a^3 z^{-1} +3 z^2 a^2+a^2 z^{-2} -5 a^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   \ -6 -5 -4 -3 -2 -1 0 χ -1             2 2 -3           1 2 1 -5         2     2 -7       1 2     1 -9     1 1       0 -11     1         1 -13 1 1           0 -15 1             1

Integral Khovanov Homology (db, data source)

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