L8a9

L8a8

L8a10

	Visit L8a9's page at Knotilus! Visit L8a9's page at the original Knot Atlas!
	L8a9 is $8_{8}^{2}$ in the Rolfsen table of links.

L8a9 Further Notes and Views

Knot presentations

Planar diagram presentation	X8192 X10,4,11,3 X16,10,7,9 X2738 X4,16,5,15 X12,5,13,6 X14,11,15,12 X6,13,1,14
Gauss code	{1, -4, 2, -5, 6, -8}, {4, -1, 3, -2, 7, -6, 8, -7, 5, -3}

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, $-{\frac {17}{48}}$)

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:L8a9/V 2,1 Data:L8a9/V 3,1 Data:L8a9/V 4,1 Data:L8a9/V 4,2 Data:L8a9/V 4,3 Data:L8a9/V 5,1 Data:L8a9/V 5,2 Data:L8a9/V 5,3 Data:L8a9/V 5,4 Data:L8a9/V 6,1 Data:L8a9/V 6,2 Data:L8a9/V 6,3 Data:L8a9/V 6,4 Data:L8a9/V 6,5 Data:L8a9/V 6,6 Data:L8a9/V 6,7 Data:L8a9/V 6,8 Data:L8a9/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 L8a9. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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, 9]]
Out[2]=	8
In[3]:=	PD[Link[8, Alternating, 9]]
Out[3]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[16, 10, 7, 9], X[2, 7, 3, 8], X[4, 16, 5, 15], X[12, 5, 13, 6], X[14, 11, 15, 12], X[6, 13, 1, 14]]
In[4]:=	GaussCode[Link[8, Alternating, 9]]
Out[4]=	GaussCode[{1, -4, 2, -5, 6, -8}, {4, -1, 3, -2, 7, -6, 8, -7, 5, -3}]
In[5]:=	BR[Link[8, Alternating, 9]]
Out[5]=	BR[Link[8, Alternating, 9]]
In[6]:=	alex = Alexander[Link[8, Alternating, 9]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, Alternating, 9]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, Alternating, 9]], KnotSignature[Link[8, Alternating, 9]]}
Out[9]=	{Infinity, -1}
In[10]:=	J=Jones[Link[8, Alternating, 9]][q]
Out[10]=	-(11/2) 3 4 6 6 6 3/2 q - ---- + ---- - ---- + ---- - ------- + 4 Sqrt[q] - 3 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[8, Alternating, 9]][q]
Out[12]=	-18 2 2 -8 2 6 8 3 - q + --- + --- + q + -- + q - q 14 10 4 q q q
In[13]:=	Kauffman[Link[8, Alternating, 9]][a, z]
Out[13]=	3 2 2 a a 2 z 3 5 z 2 2 6 2 -a + - + -- - --- - 7 a z - 7 a z - 2 a z + -- - 2 a z + a z + z z a 2 a 3 4 5 z 3 3 3 5 3 4 z 2 4 4 4 ---- + 13 a z + 13 a z + 5 a z + 3 z - -- + 8 a z + 3 a z - a 2 a 5 6 4 3 z 5 3 5 5 5 6 2 6 a z - ---- - 6 a z - 6 a z - 3 a z - 3 z - 6 a z - a 4 6 7 3 7 3 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[8, Alternating, 9]], Vassiliev[3][Link[8, Alternating, 9]]}
Out[14]=	17 {0, -(--)} 48
In[15]:=	Kh[Link[8, Alternating, 9]][q, t]
Out[15]=	3 1 2 1 2 2 4 3 4 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 q q t q t q t q t q t q t q t 3 3 2 2 2 4 2 6 3 ---- + ---- + 2 t + 2 q t + q t + 2 q t + q t 4 2 q t q t