L9a40

L9a39

L9a41

	Visit L9a40's page at Knotilus! Visit L9a40's page at the original Knot Atlas!
	L9a40 is $9_{4}^{2}$ in the Rolfsen table of links.

National swiss museum

Povray depiction

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(u^{2}v+uv^{2}+uv+u+v\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-q^{5/2}+q^{3/2}-2{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {2}{q^{11/2}}}+{\frac {1}{q^{13/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{5}-2za^{5}+z^{5}a^{3}+3z^{3}a^{3}+za^{3}+z^{5}a+4z^{3}a+4za+az^{-1}-z^{3}a^{-1}-3za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{2}+2a^{7}z^{3}+3a^{6}z^{4}-2a^{6}z^{2}+4a^{5}z^{5}-8a^{5}z^{3}+4a^{5}z+3a^{4}z^{6}-6a^{4}z^{4}+a^{4}z^{2}+2a^{3}z^{7}-4a^{3}z^{5}-2a^{3}z^{3}+2a^{3}z+a^{2}z^{8}-2a^{2}z^{6}-2a^{2}z^{4}+2a^{2}z^{2}+3az^{7}+z^{7}a^{-1}-14az^{5}-6z^{5}a^{-1}+19az^{3}+11z^{3}a^{-1}-8az+az^{-1}-6za^{-1}+a^{-1}z^{-1}+z^{8}-5z^{6}+7z^{4}-2z^{2}-1$ (db)

Vassiliev invariants

V2 and V3:

(0, ${\frac {5}{2}}$)

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:L9a40/V 2,1 Data:L9a40/V 3,1 Data:L9a40/V 4,1 Data:L9a40/V 4,2 Data:L9a40/V 4,3 Data:L9a40/V 5,1 Data:L9a40/V 5,2 Data:L9a40/V 5,3 Data:L9a40/V 5,4 Data:L9a40/V 6,1 Data:L9a40/V 6,2 Data:L9a40/V 6,3 Data:L9a40/V 6,4 Data:L9a40/V 6,5 Data:L9a40/V 6,6 Data:L9a40/V 6,7 Data:L9a40/V 6,8 Data:L9a40/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 L9a40. 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 6                   1 1 4                     0 2               2 1   1 0             1       -1 -2           3 2       1 -4         2 3         1 -6       2 1           1 -8     1 2             1 -10   1 2               -1 -12   1                 1 -14 1                   -1

Integral Khovanov Homology (db, data source)

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

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, 40]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 40]]
Out[3]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[8, 9, 1, 10], X[18, 13, 9, 14], X[14, 8, 15, 7], X[16, 6, 17, 5], X[6, 16, 7, 15], X[4, 18, 5, 17]]
In[4]:=	GaussCode[Link[9, Alternating, 40]]
Out[4]=	GaussCode[{1, -2, 3, -9, 7, -8, 6, -4}, {4, -1, 2, -3, 5, -6, 8, -7, 9, -5}]
In[5]:=	BR[Link[9, Alternating, 40]]
Out[5]=	BR[Link[9, Alternating, 40]]
In[6]:=	alex = Alexander[Link[9, Alternating, 40]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 40]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 40]], KnotSignature[Link[9, Alternating, 40]]}
Out[9]=	{Infinity, -3}
In[10]:=	J=Jones[Link[9, Alternating, 40]][q]
Out[10]=	-(13/2) 2 3 4 3 4 3 q - ----- + ---- - ---- + ---- - ---- + ------- - 2 Sqrt[q] + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 40]][q]
Out[12]=	-20 -12 2 -6 -4 -2 2 4 6 8 -q + q + -- + q + q + q + q + q + q + q 8 q
In[13]:=	Kauffman[Link[9, Alternating, 40]][a, z]
Out[13]=	1 a 6 z 3 5 2 2 2 4 2 1 - --- - - + --- + 8 a z - 2 a z - 4 a z + 2 z - 2 a z - a z + a z z a 3 6 2 8 2 11 z 3 3 3 5 3 7 3 2 a z - a z - ----- - 19 a z + 2 a z + 8 a z - 2 a z - a 5 4 2 4 4 4 6 4 6 z 5 3 5 7 z + 2 a z + 6 a z - 3 a z + ---- + 14 a z + 4 a z - a 7 5 5 6 2 6 4 6 z 7 3 7 8 4 a z + 5 z + 2 a z - 3 a z - -- - 3 a z - 2 a z - z - a 2 8 a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 40]], Vassiliev[3][Link[9, Alternating, 40]]}
Out[14]=	5 {0, -} 2
In[15]:=	Kh[Link[9, Alternating, 40]][q, t]
Out[15]=	3 3 1 1 1 2 1 2 2 -- + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 4 2 14 5 12 4 10 4 10 3 8 3 8 2 6 2 q q q t q t q t q t q t q t q t 1 2 2 t 2 2 2 3 6 4 ---- + ---- + t + --- + 2 q t + q t + q t 6 4 2 q t q t q