L6n1

L6a5

L7a1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L6n1 at Knotilus!
	L6n1 is [math]\displaystyle{ 6^3_3 }[/math] in Rolfsen's table of links. It makes three fibers in the Hopf fibration.

One modern form of the Germanic Valknut		Basic depiction		Basic symmetrical depiction		Three squares, as impossible object
Coat of arms of Suchy, Vaud, Switzerland		Rich Schwartz' "98"		Episcopal coat of arms of Dom Jacinto Bergmann (Brazil)		Heraldic badge of Admiral Lord Boyce as Lord Warden of the Cinque Ports		Canadian trade-union federation emblem.

Link Presentations

[edit Notes on L6n1's Link Presentations]

Planar diagram presentation	X6172 X12,8,9,7 X4,12,1,11 X5,11,6,10 X3845 X9,3,10,2
Gauss code	{1, 6, -5, -3}, {-4, -1, 2, 5}, {-6, 4, 3, -2}

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{w-u v}{\sqrt{u} \sqrt{v} \sqrt{w}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^4+q^2+2 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^{-2} -3 a^{-2} + a^{-4} +2-2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-2} +z^4 a^{-4} +z^3 a^{-1} +z^3 a^{-3} -4 z^2 a^{-2} -4 z^2 a^{-4} -3 z a^{-1} -3 z a^{-3} +5 a^{-2} +3 a^{-4} +3+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-2} - z^{-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   \ 0 1 2 3 4 χ 9         1 1 7         1 1 5     1     1 3 1         1 1 3 1       2 -1 2         2

Integral Khovanov Homology (db, data source)

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

L6a5 L7a1