L9a36

L9a35

L9a37

	Visit L9a36's page at Knotilus! Visit L9a36's page at the original Knot Atlas!
	L9a36 is $9_{1}^{2}$ in the Rolfsen table of links.

L9a36 Further Notes and Views

Knot presentations

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X18,12,9,11 X14,6,15,5 X16,8,17,7 X2,9,3,10 X4,14,5,13 X6,16,7,15 X8,18,1,17
Gauss code	{1, -6, 2, -7, 4, -8, 5, -9}, {6, -1, 3, -2, 7, -4, 8, -5, 9, -3}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{3}v^{3}-u^{3}v^{2}-u^{2}v^{3}+u^{2}v^{2}-u^{2}v-uv^{2}+uv-u-v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-3q^{9/2}+2q^{7/2}-2q^{5/2}+q^{3/2}+q^{19/2}-2q^{17/2}+2q^{15/2}-3q^{13/2}+3q^{11/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$z^{5}a^{-7}+4z^{3}a^{-7}+3za^{-7}-z^{7}a^{-5}-6z^{5}a^{-5}-11z^{3}a^{-5}-7za^{-5}-a^{-5}z^{-1}+z^{5}a^{-3}+5z^{3}a^{-3}+6za^{-3}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-4}-z^{8}a^{-6}-z^{7}a^{-3}-3z^{7}a^{-5}-2z^{7}a^{-7}+5z^{6}a^{-4}+3z^{6}a^{-6}-2z^{6}a^{-8}+6z^{5}a^{-3}+15z^{5}a^{-5}+7z^{5}a^{-7}-2z^{5}a^{-9}-6z^{4}a^{-4}+4z^{4}a^{-8}-2z^{4}a^{-10}-11z^{3}a^{-3}-21z^{3}a^{-5}-6z^{3}a^{-7}+2z^{3}a^{-9}-2z^{3}a^{-11}-2z^{2}a^{-6}+z^{2}a^{-10}-z^{2}a^{-12}+7za^{-3}+9za^{-5}+2za^{-7}+za^{-9}+za^{-11}+a^{-4}-a^{-3}z^{-1}-a^{-5}z^{-1}$ (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:L9a36/V 2,1 Data:L9a36/V 3,1 Data:L9a36/V 4,1 Data:L9a36/V 4,2 Data:L9a36/V 4,3 Data:L9a36/V 5,1 Data:L9a36/V 5,2 Data:L9a36/V 5,3 Data:L9a36/V 5,4 Data:L9a36/V 6,1 Data:L9a36/V 6,2 Data:L9a36/V 6,3 Data:L9a36/V 6,4 Data:L9a36/V 6,5 Data:L9a36/V 6,6 Data:L9a36/V 6,7 Data:L9a36/V 6,8 Data:L9a36/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=$5 is the signature of L9a36. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 χ 20                   1 -1 18                 1   1 16               1 1   0 14             2 1     1 12           2 2       0 10         1 1         0 8       1 2           1 6     1 1             0 4   1 2               1 2                     0 0 1                   1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=4$ $i=6$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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, 36]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 36]]
Out[3]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[18, 12, 9, 11], X[14, 6, 15, 5], X[16, 8, 17, 7], X[2, 9, 3, 10], X[4, 14, 5, 13], X[6, 16, 7, 15], X[8, 18, 1, 17]]
In[4]:=	GaussCode[Link[9, Alternating, 36]]
Out[4]=	GaussCode[{1, -6, 2, -7, 4, -8, 5, -9}, {6, -1, 3, -2, 7, -4, 8, -5, 9, -3}]
In[5]:=	BR[Link[9, Alternating, 36]]
Out[5]=	BR[Link[9, Alternating, 36]]
In[6]:=	alex = Alexander[Link[9, Alternating, 36]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 36]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 36]], KnotSignature[Link[9, Alternating, 36]]}
Out[9]=	{Infinity, 5}
In[10]:=	J=Jones[Link[9, Alternating, 36]][q]
Out[10]=	3/2 5/2 7/2 9/2 11/2 13/2 -Sqrt[q] + q - 2 q + 2 q - 3 q + 3 q - 3 q + 15/2 17/2 19/2 2 q - 2 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 36]][q]
Out[12]=	2 4 6 8 12 16 18 20 22 28 q + q + q + q + q + 2 q + q + q + q - q
In[13]:=	Kauffman[Link[9, Alternating, 36]][a, z]
Out[13]=	2 2 2 -4 1 1 z z 2 z 9 z 7 z z z 2 z a - ---- - ---- + --- + -- + --- + --- + --- - --- + --- - ---- - 5 3 11 9 7 5 3 12 10 6 a z a z a a a a a a a a 3 3 3 3 3 4 4 4 5 2 z 2 z 6 z 21 z 11 z 2 z 4 z 6 z 2 z ---- + ---- - ---- - ----- - ----- - ---- + ---- - ---- - ---- + 11 9 7 5 3 10 8 4 9 a a a a a a a a a 5 5 5 6 6 6 7 7 7 8 8 7 z 15 z 6 z 2 z 3 z 5 z 2 z 3 z z z z ---- + ----- + ---- - ---- + ---- + ---- - ---- - ---- - -- - -- - -- 7 5 3 8 6 4 7 5 3 6 4 a a a a a a a a a a a
In[14]:=	{Vassiliev[2][Link[9, Alternating, 36]], Vassiliev[3][Link[9, Alternating, 36]]}
Out[14]=	41 {0, --} 24
In[15]:=	Kh[Link[9, Alternating, 36]][q, t]
Out[15]=	4 4 6 -2 q 6 8 8 2 10 2 10 3 2 q + q + t + -- + q t + q t + 2 q t + q t + q t + t 12 3 12 4 14 4 14 5 16 5 16 6 18 6 2 q t + 2 q t + 2 q t + q t + q t + q t + q t + 20 7 q t