L6a2

L6a1

L6a3

	Visit L6a2's page at Knotilus! Visit L6a2's page at the original Knot Atlas!
	L6a2 is $6_{2}^{2}$ 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 $u$, $v$, $w$, ...)	${\frac {u^{2}(-v)-uv^{2}+uv-u-v}{uv}}$ (db)
Jones polynomial	$-{\frac {1}{q^{3/2}}}+{\frac {1}{q^{5/2}}}-{\frac {2}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {2}{q^{11/2}}}+{\frac {1}{q^{13/2}}}-{\frac {1}{q^{15/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z+a^{7}z^{-1}-a^{5}z^{3}-2a^{5}z-a^{5}z^{-1}-a^{3}z^{3}-2a^{3}z$ (db)
Kauffman polynomial	$-z^{3}a^{9}+2za^{9}-z^{4}a^{8}+z^{2}a^{8}-z^{5}a^{7}+2z^{3}a^{7}-3za^{7}+a^{7}z^{-1}-2z^{4}a^{6}+2z^{2}a^{6}-a^{6}-z^{5}a^{5}+2z^{3}a^{5}-3za^{5}+a^{5}z^{-1}-z^{4}a^{4}+z^{2}a^{4}-z^{3}a^{3}+2za^{3}$ (db)

Vassiliev invariants

V2 and V3:

(0, $-{\frac {161}{16}}$)

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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$-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)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-6$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

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