L6a3

L6a2

L6a4

	Visit L6a3's page at Knotilus! Visit L6a3's page at the original Knot Atlas!
	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.

Knot 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}

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)

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ -\frac{177}{16} }[/math])

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 Data:L6a3/V 2,1 Data:L6a3/V 3,1 Data:L6a3/V 4,1 Data:L6a3/V 4,2 Data:L6a3/V 4,3 Data:L6a3/V 5,1 Data:L6a3/V 5,2 Data:L6a3/V 5,3 Data:L6a3/V 5,4 Data:L6a3/V 6,1 Data:L6a3/V 6,2 Data:L6a3/V 6,3 Data:L6a3/V 6,4 Data:L6a3/V 6,5 Data:L6a3/V 6,6 Data:L6a3/V 6,7 Data:L6a3/V 6,8 Data:L6a3/V 6,9

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

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]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]-5 is the signature of L6a3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   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.

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math]

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Link[6, Alternating, 3]]
Out[2]=	6
In[3]:=	PD[Link[6, Alternating, 3]]
Out[3]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[12, 5, 7, 6], X[6, 7, 1, 8], X[4, 11, 5, 12]]
In[4]:=	GaussCode[Link[6, Alternating, 3]]
Out[4]=	GaussCode[{1, -2, 3, -6, 4, -5}, {5, -1, 2, -3, 6, -4}]
In[5]:=	BR[Link[6, Alternating, 3]]
Out[5]=	BR[Link[6, Alternating, 3]]
In[6]:=	alex = Alexander[Link[6, Alternating, 3]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[6, Alternating, 3]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[6, Alternating, 3]], KnotSignature[Link[6, Alternating, 3]]}
Out[9]=	{Infinity, -5}
In[10]:=	J=Jones[Link[6, Alternating, 3]][q]
Out[10]=	-(17/2) -(15/2) -(13/2) -(11/2) -(9/2) -(5/2) -q + q - q + q - q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[6, Alternating, 3]][q]
Out[12]=	-26 -24 -22 -16 -14 2 -10 -8 q + q + q + q + q + --- + q + q 12 q
In[13]:=	Kauffman[Link[6, Alternating, 3]][a, z]
Out[13]=	5 7 6 a a 5 7 9 11 6 2 8 2 -a + -- + -- - 6 a z - 4 a z + a z - a z + 3 a z + 2 a z - z z 10 2 5 3 7 3 9 3 6 4 8 4 5 5 7 5 a z + 5 a z + 4 a z - a z - a z - a z - a z - a z
In[14]:=	{Vassiliev[2][Link[6, Alternating, 3]], Vassiliev[3][Link[6, Alternating, 3]]}
Out[14]=	177 {0, -(---)} 16
In[15]:=	Kh[Link[6, Alternating, 3]][q, t]
Out[15]=	-6 -4 1 1 1 1 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ----- 18 6 16 6 16 5 12 4 12 3 8 2 q t q t q t q t q t q t