L9a9

L9a8

L9a10

	Visit L9a9's page at Knotilus! Visit L9a9's page at the original Knot Atlas!
	L9a9 is $9_{37}^{2}$ in the Rolfsen table of links.

L9a9 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X14,7,15,8 X16,9,17,10 X8,15,9,16 X4,17,1,18 X12,6,13,5 X10,4,11,3 X18,12,5,11 X2,14,3,13
Gauss code	{1, -9, 7, -5}, {6, -1, 2, -4, 3, -7, 8, -6, 9, -2, 4, -3, 5, -8}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}+1\right)\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {3}{q^{9/2}}}-q^{7/2}+{\frac {5}{q^{7/2}}}+3q^{5/2}-{\frac {7}{q^{5/2}}}-5q^{3/2}+{\frac {8}{q^{3/2}}}+{\frac {1}{q^{11/2}}}+6{\sqrt {q}}-{\frac {9}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-3a^{3}z^{3}-2a^{3}z+az^{7}+5az^{5}-z^{5}a^{-1}+8az^{3}-3z^{3}a^{-1}+4az-2za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{8}-2z^{8}-4a^{3}z^{7}-8az^{7}-4z^{7}a^{-1}-4a^{4}z^{6}-3z^{6}a^{-2}+z^{6}-3a^{5}z^{5}+7a^{3}z^{5}+22az^{5}+11z^{5}a^{-1}-z^{5}a^{-3}-a^{6}z^{4}+5a^{4}z^{4}+3a^{2}z^{4}+7z^{4}a^{-2}+4z^{4}+4a^{5}z^{3}-8a^{3}z^{3}-24az^{3}-10z^{3}a^{-1}+2z^{3}a^{-3}+a^{6}z^{2}-a^{4}z^{2}-3a^{2}z^{2}-2z^{2}a^{-2}-3z^{2}+4a^{3}z+8az+4za^{-1}+1-az^{-1}-a^{-1}z^{-1}$ (db)

Vassiliev invariants

V2 and V3:

(0, 1)

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:L9a9/V 2,1 Data:L9a9/V 3,1 Data:L9a9/V 4,1 Data:L9a9/V 4,2 Data:L9a9/V 4,3 Data:L9a9/V 5,1 Data:L9a9/V 5,2 Data:L9a9/V 5,3 Data:L9a9/V 5,4 Data:L9a9/V 6,1 Data:L9a9/V 6,2 Data:L9a9/V 6,3 Data:L9a9/V 6,4 Data:L9a9/V 6,5 Data:L9a9/V 6,6 Data:L9a9/V 6,7 Data:L9a9/V 6,8 Data:L9a9/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 L9a9. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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