L9a11

L9a10

L9a12

	Visit L9a11's page at Knotilus! Visit L9a11's page at the original Knot Atlas!
	L9a11 is $9_{26}^{2}$ in the Rolfsen table of links.

L9a11 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X12,3,13,4 X14,8,15,7 X18,16,5,15 X16,9,17,10 X8,17,9,18 X10,14,11,13 X2536 X4,11,1,12
Gauss code	{1, -8, 2, -9}, {8, -1, 3, -6, 5, -7, 9, -2, 7, -3, 4, -5, 6, -4}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{3}-4uv^{2}+6uv-2u-2v^{3}+6v^{2}-4v+1}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {5}{q^{9/2}}}+{\frac {7}{q^{7/2}}}+q^{5/2}-{\frac {9}{q^{5/2}}}-4q^{3/2}+{\frac {9}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}+6{\sqrt {q}}-{\frac {8}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}-3a^{5}z-2a^{5}z^{-1}+3a^{3}z^{3}+4a^{3}z+2a^{3}z^{-1}-az^{5}-2az^{3}+z^{3}a^{-1}-3az-az^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{8}-a^{2}z^{8}-2a^{5}z^{7}-6a^{3}z^{7}-4az^{7}-2a^{6}z^{6}-5a^{4}z^{6}-9a^{2}z^{6}-6z^{6}-a^{7}z^{5}+5a^{3}z^{5}-4z^{5}a^{-1}+4a^{6}z^{4}+12a^{4}z^{4}+17a^{2}z^{4}-z^{4}a^{-2}+8z^{4}+3a^{7}z^{3}+7a^{5}z^{3}+7a^{3}z^{3}+7az^{3}+4z^{3}a^{-1}-2a^{6}z^{2}-7a^{4}z^{2}-7a^{2}z^{2}-2z^{2}-3a^{7}z-7a^{5}z-8a^{3}z-4az+a^{4}+a^{7}z^{-1}+2a^{5}z^{-1}+2a^{3}z^{-1}+az^{-1}$ (db)

Vassiliev invariants

V2 and V3:

(0, $-{\frac {137}{24}}$)

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:L9a11/V 2,1 Data:L9a11/V 3,1 Data:L9a11/V 4,1 Data:L9a11/V 4,2 Data:L9a11/V 4,3 Data:L9a11/V 5,1 Data:L9a11/V 5,2 Data:L9a11/V 5,3 Data:L9a11/V 5,4 Data:L9a11/V 6,1 Data:L9a11/V 6,2 Data:L9a11/V 6,3 Data:L9a11/V 6,4 Data:L9a11/V 6,5 Data:L9a11/V 6,6 Data:L9a11/V 6,7 Data:L9a11/V 6,8 Data:L9a11/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 L9a11. 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           5 4       -1 -4         4 4         0 -6       3 5           2 -8     2 4             -2 -10   1 4               3 -12   1                 -1 -14 1                   1

Integral Khovanov Homology (db, data source)

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

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, 11]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 11]]
Out[3]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[14, 8, 15, 7], X[18, 16, 5, 15], X[16, 9, 17, 10], X[8, 17, 9, 18], X[10, 14, 11, 13], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[4]:=	GaussCode[Link[9, Alternating, 11]]
Out[4]=	GaussCode[{1, -8, 2, -9}, {8, -1, 3, -6, 5, -7, 9, -2, 7, -3, 4, -5, 6, -4}]
In[5]:=	BR[Link[9, Alternating, 11]]
Out[5]=	BR[Link[9, Alternating, 11]]
In[6]:=	alex = Alexander[Link[9, Alternating, 11]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 11]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 11]], KnotSignature[Link[9, Alternating, 11]]}
Out[9]=	{Infinity, -1}
In[10]:=	J=Jones[Link[9, Alternating, 11]][q]
Out[10]=	-(13/2) 2 5 7 9 9 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, 11]][q]
Out[12]=	-22 2 -16 2 2 -10 2 -2 2 6 8 3 + q + --- + q + --- - --- + q + -- - q - q + 2 q - q 20 14 12 4 q q q q
In[13]:=	Kauffman[Link[9, Alternating, 11]][a, z]
Out[13]=	3 5 7 4 a 2 a 2 a a 3 5 7 2 a + - + ---- + ---- + -- - 4 a z - 8 a z - 7 a z - 3 a z - 2 z - z z z z 3 2 2 4 2 6 2 4 z 3 3 3 5 3 7 a z - 7 a z - 2 a z + ---- + 7 a z + 7 a z + 7 a z + a 4 5 7 3 4 z 2 4 4 4 6 4 4 z 3 a z + 8 z - -- + 17 a z + 12 a z + 4 a z - ---- + 2 a a 3 5 7 5 6 2 6 4 6 6 6 7 5 a z - a z - 6 z - 9 a z - 5 a z - 2 a z - 4 a z - 3 7 5 7 2 8 4 8 6 a z - 2 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 11]], Vassiliev[3][Link[9, Alternating, 11]]}
Out[14]=	137 {0, -(---)} 24
In[15]:=	Kh[Link[9, Alternating, 11]][q, t]
Out[15]=	4 1 1 1 4 2 4 3 5 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 q q t q t q t q t q t q t q t 5 4 4 5 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