L6a2

L6a1

L6a3

	Visit L6a2's page at Knotilus! Visit L6a2's page at the original Knot Atlas!
	L6a2 is [math]\displaystyle{ 6^2_2 }[/math] in the Rolfsen table of links.

Mongolian ornament (4 crossings are unnecessary)

Knot presentations

Planar diagram presentation	X8192 X12,5,7,6 X10,3,11,4 X4,11,5,12 X2738 X6,9,1,10
Gauss code	{1, -5, 3, -4, 2, -6}, {5, -1, 6, -3, 4, -2}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ \frac{u^2 (-v)-u v^2+u v-u-v}{u v} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{2}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{1}{q^{13/2}}-\frac{1}{q^{15/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z+a^7 z^{-1} -a^5 z^3-2 a^5 z-a^5 z^{-1} -a^3 z^3-2 a^3 z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^3 a^9+2 z a^9-z^4 a^8+z^2 a^8-z^5 a^7+2 z^3 a^7-3 z a^7+a^7 z^{-1} -2 z^4 a^6+2 z^2 a^6-a^6-z^5 a^5+2 z^3 a^5-3 z a^5+a^5 z^{-1} -z^4 a^4+z^2 a^4-z^3 a^3+2 z a^3 }[/math] (db)

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ -\frac{161}{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:L6a2/V 2,1 Data:L6a2/V 3,1 Data:L6a2/V 4,1 Data:L6a2/V 4,2 Data:L6a2/V 4,3 Data:L6a2/V 5,1 Data:L6a2/V 5,2 Data:L6a2/V 5,3 Data:L6a2/V 5,4 Data:L6a2/V 6,1 Data:L6a2/V 6,2 Data:L6a2/V 6,3 Data:L6a2/V 6,4 Data:L6a2/V 6,5 Data:L6a2/V 6,6 Data:L6a2/V 6,7 Data:L6a2/V 6,8 Data:L6a2/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]-3 is the signature of L6a2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 χ -2             1 1 -4           1 1 0 -6         1     1 -8       1 1     0 -10     1 1       0 -12     1         1 -14 1 1           0 -16 1             1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_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} }[/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, 2]]
Out[2]=	6
In[3]:=	PD[Link[6, Alternating, 2]]
Out[3]=	PD[X[8, 1, 9, 2], X[12, 5, 7, 6], X[10, 3, 11, 4], X[4, 11, 5, 12], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[4]:=	GaussCode[Link[6, Alternating, 2]]
Out[4]=	GaussCode[{1, -5, 3, -4, 2, -6}, {5, -1, 6, -3, 4, -2}]
In[5]:=	BR[Link[6, Alternating, 2]]
Out[5]=	BR[Link[6, Alternating, 2]]
In[6]:=	alex = Alexander[Link[6, Alternating, 2]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[6, Alternating, 2]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[6, Alternating, 2]], KnotSignature[Link[6, Alternating, 2]]}
Out[9]=	{Infinity, -3}
In[10]:=	J=Jones[Link[6, Alternating, 2]][q]
Out[10]=	-(15/2) -(13/2) 2 2 2 -(5/2) -(3/2) -q + q - ----- + ---- - ---- + q - q 11/2 9/2 7/2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[6, Alternating, 2]][q]
Out[12]=	-24 -22 -20 2 -16 -14 -8 -4 q + q + q + --- + q + q + q + q 18 q
In[13]:=	Kauffman[Link[6, Alternating, 2]][a, z]
Out[13]=	5 7 6 a a 3 5 7 9 4 2 6 2 -a + -- + -- + 2 a z - 3 a z - 3 a z + 2 a z + a z + 2 a z + z z 8 2 3 3 5 3 7 3 9 3 4 4 6 4 8 4 a z - a z + 2 a z + 2 a z - a z - a z - 2 a z - a z - 5 5 7 5 a z - a z
In[14]:=	{Vassiliev[2][Link[6, Alternating, 2]], Vassiliev[3][Link[6, Alternating, 2]]}
Out[14]=	161 {0, -(---)} 16
In[15]:=	Kh[Link[6, Alternating, 2]][q, t]
Out[15]=	-4 -2 1 1 1 1 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 16 6 14 6 14 5 12 4 10 4 10 3 q t q t q t q t q t q t 1 1 1 1 ----- + ----- + ----- + ---- 8 3 8 2 6 2 4 q t q t q t q t