L9a22

L9a21

L9a23

	Visit L9a22's page at Knotilus! Visit L9a22's page at the original Knot Atlas!
	L9a22 is $9_{35}^{2}$ in the Rolfsen table of links.

L9a22 Further Notes and Views

Knot presentations

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X16,12,17,11 X12,6,13,5 X4,17,5,18 X14,7,15,8 X18,13,7,14 X6,16,1,15
Gauss code	{1, -2, 3, -6, 5, -9}, {7, -1, 2, -3, 4, -5, 8, -7, 9, -4, 6, -8}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{4}-2u^{2}v^{3}+2u^{2}v^{2}-u^{2}v-uv^{4}+3uv^{3}-3uv^{2}+3uv-u-v^{3}+2v^{2}-2v+1}{uv^{2}}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {7}{q^{5/2}}}-{\frac {8}{q^{7/2}}}+{\frac {7}{q^{9/2}}}-{\frac {5}{q^{11/2}}}+{\frac {3}{q^{13/2}}}-{\frac {1}{q^{15/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{3}z^{7}+a^{5}z^{5}-5a^{3}z^{5}+az^{5}+3a^{5}z^{3}-8a^{3}z^{3}+3az^{3}+2a^{5}z-4a^{3}z+az+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{9}-3z^{4}a^{8}+z^{2}a^{8}-5z^{5}a^{7}+4z^{3}a^{7}-za^{7}-6z^{6}a^{6}+8z^{4}a^{6}-3z^{2}a^{6}-5z^{7}a^{5}+8z^{5}a^{5}-3z^{3}a^{5}+za^{5}-2z^{8}a^{4}-2z^{6}a^{4}+13z^{4}a^{4}-7z^{2}a^{4}-8z^{7}a^{3}+24z^{5}a^{3}-19z^{3}a^{3}+4za^{3}+a^{3}z^{-1}-2z^{8}a^{2}+3z^{6}a^{2}+5z^{4}a^{2}-5z^{2}a^{2}-a^{2}-3z^{7}a+11z^{5}a-11z^{3}a+2za+az^{-1}-z^{6}+3z^{4}-2z^{2}$ (db)

Vassiliev invariants

V2 and V3:

(0, $-{\frac {65}{48}}$)

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:L9a22/V 2,1 Data:L9a22/V 3,1 Data:L9a22/V 4,1 Data:L9a22/V 4,2 Data:L9a22/V 4,3 Data:L9a22/V 5,1 Data:L9a22/V 5,2 Data:L9a22/V 5,3 Data:L9a22/V 5,4 Data:L9a22/V 6,1 Data:L9a22/V 6,2 Data:L9a22/V 6,3 Data:L9a22/V 6,4 Data:L9a22/V 6,5 Data:L9a22/V 6,6 Data:L9a22/V 6,7 Data:L9a22/V 6,8 Data:L9a22/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 L9a22. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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[9, Alternating, 22]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 22]]
Out[3]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[16, 12, 17, 11], X[12, 6, 13, 5], X[4, 17, 5, 18], X[14, 7, 15, 8], X[18, 13, 7, 14], X[6, 16, 1, 15]]
In[4]:=	GaussCode[Link[9, Alternating, 22]]
Out[4]=	GaussCode[{1, -2, 3, -6, 5, -9}, {7, -1, 2, -3, 4, -5, 8, -7, 9, -4, 6, -8}]
In[5]:=	BR[Link[9, Alternating, 22]]
Out[5]=	BR[Link[9, Alternating, 22]]
In[6]:=	alex = Alexander[Link[9, Alternating, 22]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 22]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 22]], KnotSignature[Link[9, Alternating, 22]]}
Out[9]=	{Infinity, -3}
In[10]:=	J=Jones[Link[9, Alternating, 22]][q]
Out[10]=	-(15/2) 3 5 7 8 7 7 4 -q + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 3 Sqrt[q] + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 22]][q]
Out[12]=	-22 -20 -18 -16 -14 -12 -10 4 -6 3 2 q - q + q - q - q + q - q + -- + q + -- + -- + 8 4 2 q q q 2 4 q - q
In[13]:=	Kauffman[Link[9, Alternating, 22]][a, z]
Out[13]=	3 2 a a 3 5 7 2 2 2 -a + - + -- + 2 a z + 4 a z + a z - a z - 2 z - 5 a z - z z 4 2 6 2 8 2 3 3 3 5 3 7 3 7 a z - 3 a z + a z - 11 a z - 19 a z - 3 a z + 4 a z - 9 3 4 2 4 4 4 6 4 8 4 5 a z + 3 z + 5 a z + 13 a z + 8 a z - 3 a z + 11 a z + 3 5 5 5 7 5 6 2 6 4 6 6 6 24 a z + 8 a z - 5 a z - z + 3 a z - 2 a z - 6 a z - 7 3 7 5 7 2 8 4 8 3 a z - 8 a z - 5 a z - 2 a z - 2 a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 22]], Vassiliev[3][Link[9, Alternating, 22]]}
Out[14]=	65 {0, -(--)} 48
In[15]:=	Kh[Link[9, Alternating, 22]][q, t]
Out[15]=	3 5 1 2 1 3 2 4 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 4 2 16 6 14 5 12 5 12 4 10 4 10 3 8 3 q q q t q t q t q t q t q t q t 4 5 4 3 2 t 2 2 2 4 3 ----- + ----- + ---- + ---- + 2 t + --- + t + 2 q t + q t 8 2 6 2 6 4 2 q t q t q t q t q