L9a35

L9a34

L9a36

	Visit L9a35's page at Knotilus! Visit L9a35's page at the original Knot Atlas!
	L9a35 is $9_{9}^{2}$ in the Rolfsen table of links.

More symmetric depiction

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)t(1)^{2}+t(2)^{2}t(1)-t(2)t(1)+t(1)+t(2)\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{9/2}+{\frac {1}{q^{9/2}}}+3q^{7/2}-{\frac {2}{q^{7/2}}}-5q^{5/2}+{\frac {3}{q^{5/2}}}+6q^{3/2}-{\frac {6}{q^{3/2}}}-7{\sqrt {q}}+{\frac {6}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$az^{5}+z^{5}a^{-1}-a^{3}z^{3}+3az^{3}+2z^{3}a^{-1}-z^{3}a^{-3}-2a^{3}z+3az-za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{8}-z^{8}-2a^{3}z^{7}-5az^{7}-3z^{7}a^{-1}-a^{4}z^{6}-5z^{6}a^{-2}-4z^{6}+8a^{3}z^{5}+14az^{5}+z^{5}a^{-1}-5z^{5}a^{-3}+4a^{4}z^{4}+7a^{2}z^{4}+6z^{4}a^{-2}-3z^{4}a^{-4}+12z^{4}-10a^{3}z^{3}-12az^{3}+4z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}-4a^{4}z^{2}-6a^{2}z^{2}-2z^{2}a^{-2}+z^{2}a^{-4}-5z^{2}+4a^{3}z+6az-2za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (db)

Vassiliev invariants

V2 and V3:

(0, ${\frac {3}{2}}$)

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:L9a35/V 2,1 Data:L9a35/V 3,1 Data:L9a35/V 4,1 Data:L9a35/V 4,2 Data:L9a35/V 4,3 Data:L9a35/V 5,1 Data:L9a35/V 5,2 Data:L9a35/V 5,3 Data:L9a35/V 5,4 Data:L9a35/V 6,1 Data:L9a35/V 6,2 Data:L9a35/V 6,3 Data:L9a35/V 6,4 Data:L9a35/V 6,5 Data:L9a35/V 6,6 Data:L9a35/V 6,7 Data:L9a35/V 6,8 Data:L9a35/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 L9a35. 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 10                   1 1 8                 2   -2 6               3 1   2 4             3 2     -1 2           4 3       1 0         4 5         1 -2       2 2           0 -4     1 4             3 -6   1 2               -1 -8   1                 1 -10 1                   -1

Integral Khovanov Homology (db, data source)

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

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, 35]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 35]]
Out[3]=	PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[16, 8, 17, 7], X[14, 6, 15, 5], X[18, 13, 9, 14], X[6, 16, 7, 15], X[4, 18, 5, 17], X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[4]:=	GaussCode[Link[9, Alternating, 35]]
Out[4]=	GaussCode[{1, -8, 2, -7, 4, -6, 3, -9}, {8, -1, 9, -2, 5, -4, 6, -3, 7, -5}]
In[5]:=	BR[Link[9, Alternating, 35]]
Out[5]=	BR[Link[9, Alternating, 35]]
In[6]:=	alex = Alexander[Link[9, Alternating, 35]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 35]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 35]], KnotSignature[Link[9, Alternating, 35]]}
Out[9]=	{Infinity, 1}
In[10]:=	J=Jones[Link[9, Alternating, 35]][q]
Out[10]=	-(9/2) 2 3 6 6 3/2 5/2 q - ---- + ---- - ---- + ------- - 7 Sqrt[q] + 6 q - 5 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 3 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 35]][q]
Out[12]=	-14 3 -4 2 4 6 8 12 14 2 - q + -- + q + -- + 2 q - q + q - q + q 6 2 q q
In[13]:=	Kauffman[Link[9, Alternating, 35]][a, z]
Out[13]=	2 2 1 a 2 z 3 2 z 2 z 2 2 1 - --- - - - --- + 6 a z + 4 a z - 5 z + -- - ---- - 6 a z - a z z 3 4 2 a a a 3 3 3 4 4 2 z 5 z 4 z 3 3 3 4 3 z 4 a z - -- + ---- + ---- - 12 a z - 10 a z + 12 z - ---- + 5 3 a 4 a a a 4 5 5 6 z 2 4 4 4 5 z z 5 3 5 6 ---- + 7 a z + 4 a z - ---- + -- + 14 a z + 8 a z - 4 z - 2 3 a a a 6 7 5 z 4 6 3 z 7 3 7 8 2 8 ---- - a z - ---- - 5 a z - 2 a z - z - a z 2 a a
In[14]:=	{Vassiliev[2][Link[9, Alternating, 35]], Vassiliev[3][Link[9, Alternating, 35]]}
Out[14]=	3 {0, -} 2
In[15]:=	Kh[Link[9, Alternating, 35]][q, t]
Out[15]=	2 1 1 1 2 1 4 2 4 5 + 4 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t q t q t q t q t q t q t q t 2 2 4 4 2 6 2 6 3 8 3 10 4 ---- + 3 q t + 3 q t + 2 q t + 3 q t + q t + 2 q t + q t 2 q t