L8n7

L8n6

L8n8

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n7 at Knotilus!
	L8n7 is [math]\displaystyle{ 8^4_{2} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L8n7's Link Presentations]

Planar diagram presentation	X6172 X2536 X11,13,12,16 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X15,9,16,12
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 8}, {-7, 6, -8, 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{t(1) t(2)-t(1) t(3) t(2)+t(3) t(2)-t(1) t(4) t(2)-t(3)+t(1) t(4)+t(3) t(4)-t(4)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{13/2}+q^{11/2}-4 q^{9/2}+q^{7/2}-4 q^{5/2}+2 q^{3/2}-3 \sqrt{q} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ - a^{-7} z^{-3} - a^{-7} z^{-1} +3 a^{-5} z^{-3} +3 z a^{-5} +5 a^{-5} z^{-1} -2 z^3 a^{-3} -3 a^{-3} z^{-3} -6 z a^{-3} -7 a^{-3} z^{-1} + a^{-1} z^{-3} +3 z a^{-1} +3 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-7} -4 z^3 a^{-7} + a^{-7} z^{-3} +6 z a^{-7} -4 a^{-7} z^{-1} +z^6 a^{-6} -z^4 a^{-6} -6 z^2 a^{-6} -3 a^{-6} z^{-2} +8 a^{-6} +5 z^5 a^{-5} -16 z^3 a^{-5} +3 a^{-5} z^{-3} +14 z a^{-5} -9 a^{-5} z^{-1} +z^6 a^{-4} +2 z^4 a^{-4} -12 z^2 a^{-4} -6 a^{-4} z^{-2} +15 a^{-4} +4 z^5 a^{-3} -12 z^3 a^{-3} +3 a^{-3} z^{-3} +14 z a^{-3} -9 a^{-3} z^{-1} +3 z^4 a^{-2} -6 z^2 a^{-2} -3 a^{-2} z^{-2} +8 a^{-2} + a^{-1} z^{-3} +6 z a^{-1} -4 a^{-1} 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   \ 0 1 2 3 4 5 6 χ 14             1 1 12               0 10         4 1   3 8       1 4     3 6     3         3 4 1   1         2 2 4 3           1 0 3             3

Integral Khovanov Homology (db, data source)

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