L7a6

L7a5

L7a7

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7a6 at Knotilus!
	L7a6 is [math]\displaystyle{ 7^2_1 }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L7a6's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X14,10,7,9 X12,6,13,5 X2738 X4,12,5,11 X6,14,1,13
Gauss code	{1, -5, 2, -6, 4, -7}, {5, -1, 3, -2, 6, -4, 7, -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(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+t(2) t(1)-t(1)-t(2)+1}{t(1) t(2)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{13/2}-2 q^{11/2}+2 q^{9/2}-3 q^{7/2}+2 q^{5/2}-2 q^{3/2}+\sqrt{q}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-3} +z^3 a^{-1} -4 z^3 a^{-3} +z^3 a^{-5} +3 z a^{-1} -4 z a^{-3} +2 z a^{-5} + a^{-1} z^{-1} - a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^2 a^{-8} +2 z^3 a^{-7} -z a^{-7} +2 z^4 a^{-6} -z^2 a^{-6} +2 z^5 a^{-5} -3 z^3 a^{-5} +z a^{-5} +z^6 a^{-4} -z^4 a^{-4} -z^2 a^{-4} +3 z^5 a^{-3} -9 z^3 a^{-3} +6 z a^{-3} - a^{-3} z^{-1} +z^6 a^{-2} -3 z^4 a^{-2} +z^2 a^{-2} + a^{-2} +z^5 a^{-1} -4 z^3 a^{-1} +4 z a^{-1} - 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   \ -2 -1 0 1 2 3 4 5 χ 14               1 -1 12             1   1 10           1 1   0 8         2 1     1 6       1 2       1 4     1 1         0 2   1 2           1 0                 0 -2 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=4 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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} }[/math] [math]\displaystyle{ r=1 }[/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=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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/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.

L7a5 L7a7