L9a32

L9a31

L9a33

	Visit L9a32's page at Knotilus! Visit L9a32's page at the original Knot Atlas!
	L9a32 is $9_{40}^{2}$ in the Rolfsen table of links.

A traditional symbol of the Christian Trinity (a triquetra interlaced with a circle, or "Trinity knot")		Symmetric version, with three lines touching at center		Symmetric version, with three lines touching at cicrumference
Logo of the Canadian Undergraduate Mathematics Conference		Circled hypotrochoid

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(2)^{4}-3t(1)t(2)^{3}+3t(2)^{3}-3t(1)^{2}t(2)^{2}+5t(1)t(2)^{2}-3t(2)^{2}+3t(1)^{2}t(2)-3t(1)t(2)-t(1)^{2}}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	${\frac {7}{q^{9/2}}}-{\frac {6}{q^{7/2}}}+{\frac {3}{q^{5/2}}}-{\frac {1}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {3}{q^{19/2}}}+{\frac {5}{q^{17/2}}}-{\frac {7}{q^{15/2}}}+{\frac {8}{q^{13/2}}}-{\frac {9}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+4a^{9}z+3a^{9}z^{-1}-3a^{7}z^{3}-4a^{7}z-2a^{7}z^{-1}-3a^{5}z^{3}-3a^{5}z-a^{3}z^{3}$ (db)
Kauffman polynomial	$a^{12}z^{6}-3a^{12}z^{4}+3a^{12}z^{2}-a^{12}+3a^{11}z^{7}-10a^{11}z^{5}+9a^{11}z^{3}-2a^{11}z+a^{11}z^{-1}+2a^{10}z^{8}-a^{10}z^{6}-12a^{10}z^{4}+14a^{10}z^{2}-3a^{10}+9a^{9}z^{7}-26a^{9}z^{5}+22a^{9}z^{3}-11a^{9}z+3a^{9}z^{-1}+2a^{8}z^{8}+5a^{8}z^{6}-21a^{8}z^{4}+14a^{8}z^{2}-3a^{8}+6a^{7}z^{7}-10a^{7}z^{5}+6a^{7}z^{3}-6a^{7}z+2a^{7}z^{-1}+7a^{6}z^{6}-9a^{6}z^{4}+3a^{6}z^{2}+6a^{5}z^{5}-6a^{5}z^{3}+3a^{5}z+3a^{4}z^{4}+a^{3}z^{3}$ (db)

Vassiliev invariants

V2 and V3:

(0, $-{\frac {73}{16}}$)

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:L9a32/V 2,1 Data:L9a32/V 3,1 Data:L9a32/V 4,1 Data:L9a32/V 4,2 Data:L9a32/V 4,3 Data:L9a32/V 5,1 Data:L9a32/V 5,2 Data:L9a32/V 5,3 Data:L9a32/V 5,4 Data:L9a32/V 6,1 Data:L9a32/V 6,2 Data:L9a32/V 6,3 Data:L9a32/V 6,4 Data:L9a32/V 6,5 Data:L9a32/V 6,6 Data:L9a32/V 6,7 Data:L9a32/V 6,8 Data:L9a32/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 L9a32. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                   1 1 -4                 3 1 -2 -6               3     3 -8             4 3     -1 -10           5 3       2 -12         3 4         1 -14       4 5           -1 -16     2 4             2 -18   1 3               -2 -20   2                 2 -22 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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, 32]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 32]]
Out[3]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[18, 13, 7, 14], X[14, 9, 15, 10], X[10, 17, 11, 18], X[16, 5, 17, 6], X[2, 7, 3, 8], X[4, 11, 5, 12], X[6, 15, 1, 16]]
In[4]:=	GaussCode[Link[9, Alternating, 32]]
Out[4]=	GaussCode[{1, -7, 2, -8, 6, -9}, {7, -1, 4, -5, 8, -2, 3, -4, 9, -6, 5, -3}]
In[5]:=	BR[Link[9, Alternating, 32]]
Out[5]=	BR[Link[9, Alternating, 32]]
In[6]:=	alex = Alexander[Link[9, Alternating, 32]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 32]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 32]], KnotSignature[Link[9, Alternating, 32]]}
Out[9]=	{Infinity, -3}
In[10]:=	J=Jones[Link[9, Alternating, 32]][q]
Out[10]=	-(21/2) 3 5 7 8 9 7 6 3 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q q -(3/2) q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 32]][q]
Out[12]=	-34 2 -30 4 -22 2 -18 2 -12 2 -8 -q - --- + q + --- + q + --- + q + --- - q + --- + q - 32 24 20 14 10 q q q q q 2 -4 -- + q 6 q
In[13]:=	Kauffman[Link[9, Alternating, 32]][a, z]
Out[13]=	7 9 11 8 10 12 2 a 3 a a 5 7 9 3 a + 3 a + a - ---- - ---- - --- - 3 a z + 6 a z + 11 a z + z z z 11 6 2 8 2 10 2 12 2 3 3 2 a z - 3 a z - 14 a z - 14 a z - 3 a z - a z + 5 3 7 3 9 3 11 3 4 4 6 4 6 a z - 6 a z - 22 a z - 9 a z - 3 a z + 9 a z + 8 4 10 4 12 4 5 5 7 5 9 5 21 a z + 12 a z + 3 a z - 6 a z + 10 a z + 26 a z + 11 5 6 6 8 6 10 6 12 6 7 7 9 7 10 a z - 7 a z - 5 a z + a z - a z - 6 a z - 9 a z - 11 7 8 8 10 8 3 a z - 2 a z - 2 a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 32]], Vassiliev[3][Link[9, Alternating, 32]]}
Out[14]=	73 {0, -(--)} 16
In[15]:=	Kh[Link[9, Alternating, 32]][q, t]
Out[15]=	-4 -2 1 2 1 3 2 4 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 22 9 20 8 18 8 18 7 16 7 16 6 q t q t q t q t q t q t 4 5 3 4 5 3 4 3 ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 14 6 14 5 12 5 12 4 10 4 10 3 8 3 8 2 q t q t q t q t q t q t q t q t 3 3 ----- + ---- 6 2 4 q t q t