L6a3

L6a2

L6a4

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L6a3 at Knotilus!
	The link L6a3 is [math]\displaystyle{ 6^2_1 }[/math] in the Rolfsen table of links. It is often seen in "Magen David" (star of David) necklaces.

Ruberman, Cochran, Melvin, Akbulut, Gompf, Kirby [1]

Rich Schwartz' "72" [2]

Triangle interlaced with a circle, a traditional symbol of the Christian Trinity (less used in recent centuries)

An architectural trefoil (the outline of three overlapping circles) interlaced with an equilateral triangle, another old Christian Trinitarian symbol.

Closed granny knot.

Further images

Celtic flag proposal.	Flag of Blonay, Vaud, Switzerland.	"Eye of Providence" inside triangle interlaced with circle, Venice.	Composed of intersecting circles.
Hindu hexagram symbol with Aum/Om, from Nepal.	Double boomerang-like Kolam and Linked knot, Nagata.	A carpet.	ChatGPT bot logo.

Link Presentations

[edit Notes on L6a3's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X12,5,7,6 X6718 X4,11,5,12
Gauss code	{1, -2, 3, -6, 4, -5}, {5, -1, 2, -3, 6, -4}

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)^2 t(2)^2-t(1) t(2)-1}{t(1) t(2)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{9/2}}-\frac{1}{q^{5/2}}-\frac{1}{q^{17/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^3+3 a^7 z+a^7 z^{-1} -a^5 z^5-5 a^5 z^3-6 a^5 z-a^5 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z a^{11}-z^2 a^{10}-z^3 a^9+z a^9-z^4 a^8+2 z^2 a^8-z^5 a^7+4 z^3 a^7-4 z a^7+a^7 z^{-1} -z^4 a^6+3 z^2 a^6-a^6-z^5 a^5+5 z^3 a^5-6 z a^5+a^5 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   \ -6 -5 -4 -3 -2 -1 0 χ -4             1 1 -6             1 1 -8         1     1 -10               0 -12     1 1       0 -14               0 -16 1 1           0 -18 1             1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/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}_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.

L6a2 L6a4