L9a33

L9a32

L9a34

	Visit L9a33's page at Knotilus! Visit L9a33's page at the original Knot Atlas!
	L9a33 is $9_{24}^{2}$ in the Rolfsen table of links.

Symmetric form		Alternate symmetric version, with three lines touching at center		Alternate symmetric version, with three lines touching at circumference
Form made from 45-degree lines and circular arcs.		Depiction obtained by knotilus.		With an hypotrochoid [1].
Mexican book.

Knot presentations

Planar diagram presentation	X8192 X12,3,13,4 X18,10,7,9 X10,14,11,13 X16,5,17,6 X14,18,15,17 X2738 X4,11,5,12 X6,15,1,16
Gauss code	{1, -7, 2, -8, 5, -9}, {7, -1, 3, -4, 8, -2, 4, -6, 9, -5, 6, -3}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{2}-3u^{2}v+3u^{2}-3uv^{2}+7uv-3u+3v^{2}-3v+1}{uv}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}+{\frac {7}{q^{7/2}}}+q^{5/2}-{\frac {9}{q^{5/2}}}-4q^{3/2}+{\frac {10}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}+6{\sqrt {q}}-{\frac {8}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}-3a^{5}z-a^{5}z^{-1}+3a^{3}z^{3}+3a^{3}z-az^{5}-2az^{3}+z^{3}a^{-1}-3az$ (db)
Kauffman polynomial	$-a^{4}z^{8}-a^{2}z^{8}-3a^{5}z^{7}-7a^{3}z^{7}-4az^{7}-2a^{6}z^{6}-7a^{4}z^{6}-11a^{2}z^{6}-6z^{6}-a^{7}z^{5}+5a^{5}z^{5}+9a^{3}z^{5}-az^{5}-4z^{5}a^{-1}+3a^{6}z^{4}+18a^{4}z^{4}+24a^{2}z^{4}-z^{4}a^{-2}+8z^{4}+3a^{7}z^{3}-3a^{5}z^{3}-2a^{3}z^{3}+8az^{3}+4z^{3}a^{-1}-11a^{4}z^{2}-14a^{2}z^{2}-3z^{2}-3a^{7}z+a^{5}z+2a^{3}z-2az-a^{6}+a^{7}z^{-1}+a^{5}z^{-1}$ (db)

Vassiliev invariants

V2 and V3:

(0, $-{\frac {129}{16}}$)

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:L9a33/V 2,1 Data:L9a33/V 3,1 Data:L9a33/V 4,1 Data:L9a33/V 4,2 Data:L9a33/V 4,3 Data:L9a33/V 5,1 Data:L9a33/V 5,2 Data:L9a33/V 5,3 Data:L9a33/V 5,4 Data:L9a33/V 6,1 Data:L9a33/V 6,2 Data:L9a33/V 6,3 Data:L9a33/V 6,4 Data:L9a33/V 6,5 Data:L9a33/V 6,6 Data:L9a33/V 6,7 Data:L9a33/V 6,8 Data:L9a33/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 L9a33. 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 6                   1 -1 4                 3   3 2               3 1   -2 0             5 3     2 -2           6 4       -2 -4         3 4         -1 -6       4 6           2 -8     2 3             -1 -10     4               4 -12 1 2                 -1 -14 1                   1

Integral Khovanov Homology (db, data source)

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

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, 33]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 33]]
Out[3]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[18, 10, 7, 9], X[10, 14, 11, 13], X[16, 5, 17, 6], X[14, 18, 15, 17], X[2, 7, 3, 8], X[4, 11, 5, 12], X[6, 15, 1, 16]]
In[4]:=	GaussCode[Link[9, Alternating, 33]]
Out[4]=	GaussCode[{1, -7, 2, -8, 5, -9}, {7, -1, 3, -4, 8, -2, 4, -6, 9, -5, 6, -3}]
In[5]:=	BR[Link[9, Alternating, 33]]
Out[5]=	BR[Link[9, Alternating, 33]]
In[6]:=	alex = Alexander[Link[9, Alternating, 33]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 33]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 33]], KnotSignature[Link[9, Alternating, 33]]}
Out[9]=	{Infinity, -1}
In[10]:=	J=Jones[Link[9, Alternating, 33]][q]
Out[10]=	-(13/2) 2 6 7 9 10 8 -q + ----- - ---- + ---- - ---- + ---- - ------- + 6 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 4 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 33]][q]
Out[12]=	-22 2 2 4 2 -8 2 2 2 6 8 3 + q + --- + --- + --- + --- - q - -- - -- - q + 2 q - q 20 16 14 10 6 2 q q q q q q
In[13]:=	Kauffman[Link[9, Alternating, 33]][a, z]
Out[13]=	5 7 6 a a 3 5 7 2 2 2 -a + -- + -- - 2 a z + 2 a z + a z - 3 a z - 3 z - 14 a z - z z 3 4 4 2 4 z 3 3 3 5 3 7 3 4 z 11 a z + ---- + 8 a z - 2 a z - 3 a z + 3 a z + 8 z - -- + a 2 a 5 2 4 4 4 6 4 4 z 5 3 5 5 5 24 a z + 18 a z + 3 a z - ---- - a z + 9 a z + 5 a z - a 7 5 6 2 6 4 6 6 6 7 3 7 a z - 6 z - 11 a z - 7 a z - 2 a z - 4 a z - 7 a z - 5 7 2 8 4 8 3 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 33]], Vassiliev[3][Link[9, Alternating, 33]]}
Out[14]=	129 {0, -(---)} 16
In[15]:=	Kh[Link[9, Alternating, 33]][q, t]
Out[15]=	4 1 1 2 4 2 3 4 5 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 2 14 6 12 6 12 5 10 4 8 4 8 3 6 3 q q t q t q t q t q t q t q t 6 3 4 6 2 2 2 4 2 6 3 ----- + ----- + ---- + ---- + 3 t + 3 q t + q t + 3 q t + q t 6 2 4 2 4 2 q t q t q t q t