L9a34

L9a33

L9a35

	Visit L9a34's page at Knotilus! Visit L9a34's page at the original Knot Atlas!
	L9a34 is $9_{6}^{2}$ in the Rolfsen table of links.

L9a34 Further Notes and Views

Knot presentations

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

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, $-{\frac {29}{24}}$)

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:L9a34/V 2,1 Data:L9a34/V 3,1 Data:L9a34/V 4,1 Data:L9a34/V 4,2 Data:L9a34/V 4,3 Data:L9a34/V 5,1 Data:L9a34/V 5,2 Data:L9a34/V 5,3 Data:L9a34/V 5,4 Data:L9a34/V 6,1 Data:L9a34/V 6,2 Data:L9a34/V 6,3 Data:L9a34/V 6,4 Data:L9a34/V 6,5 Data:L9a34/V 6,6 Data:L9a34/V 6,7 Data:L9a34/V 6,8 Data:L9a34/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 L9a34. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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, 34]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 34]]
Out[3]=	PD[X[10, 1, 11, 2], X[14, 5, 15, 6], X[12, 3, 13, 4], X[16, 8, 17, 7], X[18, 15, 9, 16], X[4, 13, 5, 14], X[6, 18, 7, 17], X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[4]:=	GaussCode[Link[9, Alternating, 34]]
Out[4]=	GaussCode[{1, -8, 3, -6, 2, -7, 4, -9}, {8, -1, 9, -3, 6, -2, 5, -4, 7, -5}]
In[5]:=	BR[Link[9, Alternating, 34]]
Out[5]=	BR[Link[9, Alternating, 34]]
In[6]:=	alex = Alexander[Link[9, Alternating, 34]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 34]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 34]], KnotSignature[Link[9, Alternating, 34]]}
Out[9]=	{Infinity, -3}
In[10]:=	J=Jones[Link[9, Alternating, 34]][q]
Out[10]=	-(17/2) 2 3 6 6 6 5 4 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q 2 ------- - Sqrt[q] Sqrt[q]
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 34]][q]
Out[12]=	-26 3 -16 2 -12 2 -6 -4 2 -q + --- + q + --- + q + -- - q + q + q 18 14 8 q q q
In[13]:=	Kauffman[Link[9, Alternating, 34]][a, z]
Out[13]=	3 5 4 a a 3 5 7 9 2 2 a - -- - -- - 2 a z + 3 a z + 6 a z - a z - 2 a z - 2 a z - z z 4 2 6 2 10 2 3 3 3 5 3 7 3 a z - a z + 2 a z + 3 a z - 2 a z - 12 a z - 3 a z + 9 3 2 4 4 4 8 4 10 4 5 3 5 4 a z + 5 a z + 2 a z + 2 a z - a z - a z + 4 a z + 5 5 7 5 9 5 2 6 8 6 3 7 10 a z + 3 a z - 2 a z - 2 a z - 2 a z - 2 a z - 5 7 7 7 4 8 6 8 4 a z - 2 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 34]], Vassiliev[3][Link[9, Alternating, 34]]}
Out[14]=	29 {0, -(--)} 24
In[15]:=	Kh[Link[9, Alternating, 34]][q, t]
Out[15]=	2 3 1 1 1 2 1 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 q q q t q t q t q t q t q t 3 3 3 3 3 2 3 t 2 2 ------ + ------ + ----- + ----- + ----- + ---- + ---- + t + -- + q t 10 4 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q t q