L8a6

L8a5

L8a7

	Visit L8a6's page at Knotilus! Visit L8a6's page at the original Knot Atlas!
	L8a6 is $8_{6}^{2}$ in the Rolfsen table of links.

L8a6 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X12,3,13,4 X16,8,5,7 X14,10,15,9 X10,14,11,13 X8,16,9,15 X2536 X4,11,1,12
Gauss code	{1, -7, 2, -8}, {7, -1, 3, -6, 4, -5, 8, -2, 5, -4, 6, -3}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2uv-3u-3v+2}{{\sqrt {u}}{\sqrt {v}}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}-q^{7/2}+{\frac {1}{q^{7/2}}}+2q^{5/2}-{\frac {3}{q^{5/2}}}-2q^{3/2}+{\frac {3}{q^{3/2}}}+3{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-1}-2a^{3}z-a^{3}z^{-1}-za^{-3}+az^{3}+z^{3}a^{-1}+za^{-1}$ (db)
Kauffman polynomial	$-az^{7}-z^{7}a^{-1}-a^{2}z^{6}-2z^{6}a^{-2}-3z^{6}-a^{3}z^{5}+2az^{5}+2z^{5}a^{-1}-z^{5}a^{-3}-a^{4}z^{4}+7z^{4}a^{-2}+8z^{4}-a^{5}z^{3}-a^{3}z^{3}-3az^{3}+3z^{3}a^{-3}-5z^{2}a^{-2}-5z^{2}+2a^{5}z+2a^{3}z+az-za^{-3}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (db)

Vassiliev invariants

V2 and V3:

(0, $-{\frac {41}{24}}$)

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:L8a6/V 2,1 Data:L8a6/V 3,1 Data:L8a6/V 4,1 Data:L8a6/V 4,2 Data:L8a6/V 4,3 Data:L8a6/V 5,1 Data:L8a6/V 5,2 Data:L8a6/V 5,3 Data:L8a6/V 5,4 Data:L8a6/V 6,1 Data:L8a6/V 6,2 Data:L8a6/V 6,3 Data:L8a6/V 6,4 Data:L8a6/V 6,5 Data:L8a6/V 6,6 Data:L8a6/V 6,7 Data:L8a6/V 6,8 Data:L8a6/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 $s=$-1 is the signature of L8a6. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 χ 8                 1 1 6               1   -1 4             1 1   0 2           2 1     -1 0         2 1       1 -2       2 3         1 -4     1 1           0 -6     2             2 -8 1 1               0 -10 1                 1

Integral Khovanov Homology (db, data source)

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

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[8, Alternating, 6]]
Out[2]=	8
In[3]:=	PD[Link[8, Alternating, 6]]
Out[3]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[16, 8, 5, 7], X[14, 10, 15, 9], X[10, 14, 11, 13], X[8, 16, 9, 15], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[4]:=	GaussCode[Link[8, Alternating, 6]]
Out[4]=	GaussCode[{1, -7, 2, -8}, {7, -1, 3, -6, 4, -5, 8, -2, 5, -4, 6, -3}]
In[5]:=	BR[Link[8, Alternating, 6]]
Out[5]=	BR[Link[8, Alternating, 6]]
In[6]:=	alex = Alexander[Link[8, Alternating, 6]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, Alternating, 6]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, Alternating, 6]], KnotSignature[Link[8, Alternating, 6]]}
Out[9]=	{Infinity, -1}
In[10]:=	J=Jones[Link[8, Alternating, 6]][q]
Out[10]=	-(9/2) -(7/2) 3 3 4 3/2 -q + q - ---- + ---- - ------- + 3 Sqrt[q] - 2 q + 5/2 3/2 Sqrt[q] q q 5/2 7/2 2 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[8, Alternating, 6]][q]
Out[12]=	-16 2 -12 2 2 -4 4 8 12 1 + q + --- + q + --- + -- + q - q - q + q 14 10 8 q q q
In[13]:=	Kauffman[Link[8, Alternating, 6]][a, z]
Out[13]=	3 5 2 3 4 a a z 3 5 2 5 z 3 z a - -- - -- - -- + a z + 2 a z + 2 a z - 5 z - ---- + ---- - z z 3 2 3 a a a 4 5 5 3 3 3 5 3 4 7 z 4 4 z 2 z 5 3 a z - a z - a z + 8 z + ---- - a z - -- + ---- + 2 a z - 2 3 a a a 6 7 3 5 6 2 z 2 6 z 7 a z - 3 z - ---- - a z - -- - a z 2 a a
In[14]:=	{Vassiliev[2][Link[8, Alternating, 6]], Vassiliev[3][Link[8, Alternating, 6]]}
Out[14]=	41 {0, -(--)} 24
In[15]:=	Kh[Link[8, Alternating, 6]][q, t]
Out[15]=	3 1 1 1 2 1 1 2 2 + -- + ------ + ----- + ----- + ----- + ----- + ---- + ---- + t + 2 10 4 8 4 8 3 6 2 4 2 4 2 q q t q t q t q t q t q t q t 2 2 2 4 2 4 3 6 3 8 4 2 q t + q t + q t + q t + q t + q t