L9a12

L9a11

L9a13

	Visit L9a12's page at Knotilus! Visit L9a12's page at the original Knot Atlas!
	L9a12 is $9_{14}^{2}$ in the Rolfsen table of links.

L9a12 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-2uv^{4}+2uv^{3}-2uv^{2}+2uv-u-v^{5}+2v^{4}-2v^{3}+2v^{2}-2v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {4}{q^{9/2}}}+{\frac {2}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{23/2}}}-{\frac {2}{q^{21/2}}}+{\frac {4}{q^{19/2}}}-{\frac {5}{q^{17/2}}}+{\frac {6}{q^{15/2}}}-{\frac {7}{q^{13/2}}}+{\frac {4}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{11}(-z)-2a^{11}z^{-1}+3a^{9}z^{3}+9a^{9}z+5a^{9}z^{-1}-2a^{7}z^{5}-8a^{7}z^{3}-9a^{7}z-3a^{7}z^{-1}-a^{5}z^{5}-3a^{5}z^{3}-a^{5}z$ (db)
Kauffman polynomial	$-z^{4}a^{14}+2z^{2}a^{14}-a^{14}-2z^{5}a^{13}+3z^{3}a^{13}-2z^{6}a^{12}+z^{4}a^{12}+2z^{2}a^{12}-2z^{7}a^{11}+3z^{5}a^{11}-5z^{3}a^{11}+5za^{11}-2a^{11}z^{-1}-z^{8}a^{10}-z^{6}a^{10}+6z^{4}a^{10}-11z^{2}a^{10}+5a^{10}-5z^{7}a^{9}+17z^{5}a^{9}-26z^{3}a^{9}+16za^{9}-5a^{9}z^{-1}-z^{8}a^{8}-z^{6}a^{8}+9z^{4}a^{8}-12z^{2}a^{8}+5a^{8}-3z^{7}a^{7}+11z^{5}a^{7}-15z^{3}a^{7}+10za^{7}-3a^{7}z^{-1}-2z^{6}a^{6}+5z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+3z^{3}a^{5}-za^{5}$ (db)

Vassiliev invariants

V2 and V3:

(0, ${\frac {367}{24}}$)

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:L9a12/V 2,1 Data:L9a12/V 3,1 Data:L9a12/V 4,1 Data:L9a12/V 4,2 Data:L9a12/V 4,3 Data:L9a12/V 5,1 Data:L9a12/V 5,2 Data:L9a12/V 5,3 Data:L9a12/V 5,4 Data:L9a12/V 6,1 Data:L9a12/V 6,2 Data:L9a12/V 6,3 Data:L9a12/V 6,4 Data:L9a12/V 6,5 Data:L9a12/V 6,6 Data:L9a12/V 6,7 Data:L9a12/V 6,8 Data:L9a12/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 $s=$-5 is the signature of L9a12. 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 -4                   1 1 -6                 2 1 -1 -8               2     2 -10             2 2     0 -12           5 2       3 -14         2 3         1 -16       3 4           -1 -18     1 2             1 -20   1 3               -2 -22   1                 1 -24 1                   -1

Integral Khovanov Homology (db, data source)

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

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, 12]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 12]]
Out[3]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[18, 13, 5, 14], X[14, 7, 15, 8], X[16, 9, 17, 10], X[8, 15, 9, 16], X[10, 17, 11, 18], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[4]:=	GaussCode[Link[9, Alternating, 12]]
Out[4]=	GaussCode[{1, -8, 2, -9}, {8, -1, 4, -6, 5, -7, 9, -2, 3, -4, 6, -5, 7, -3}]
In[5]:=	BR[Link[9, Alternating, 12]]
Out[5]=	BR[Link[9, Alternating, 12]]
In[6]:=	alex = Alexander[Link[9, Alternating, 12]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 12]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 12]], KnotSignature[Link[9, Alternating, 12]]}
Out[9]=	{Infinity, -5}
In[10]:=	J=Jones[Link[9, Alternating, 12]][q]
Out[10]=	-(23/2) 2 4 5 6 7 4 4 q - ----- + ----- - ----- + ----- - ----- + ----- - ---- + 21/2 19/2 17/2 15/2 13/2 11/2 9/2 q q q q q q q 2 -(5/2) ---- - q 7/2 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 12]][q]
Out[12]=	-36 2 -32 2 -26 -24 4 2 4 2 -12 -q - --- - q - --- + q + q + --- + --- + --- + --- + q - 34 30 22 20 18 16 q q q q q q -10 -8 q + q
In[13]:=	Kauffman[Link[9, Alternating, 12]][a, z]
Out[13]=	7 9 11 8 10 14 3 a 5 a 2 a 5 7 9 5 a + 5 a - a - ---- - ---- - ----- - a z + 10 a z + 16 a z + z z z 11 6 2 8 2 10 2 12 2 14 2 5 a z - a z - 12 a z - 11 a z + 2 a z + 2 a z + 5 3 7 3 9 3 11 3 13 3 6 4 3 a z - 15 a z - 26 a z - 5 a z + 3 a z + 5 a z + 8 4 10 4 12 4 14 4 5 5 7 5 9 5 9 a z + 6 a z + a z - a z - a z + 11 a z + 17 a z + 11 5 13 5 6 6 8 6 10 6 12 6 7 7 3 a z - 2 a z - 2 a z - a z - a z - 2 a z - 3 a z - 9 7 11 7 8 8 10 8 5 a z - 2 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 12]], Vassiliev[3][Link[9, Alternating, 12]]}
Out[14]=	367 {0, ---} 24
In[15]:=	Kh[Link[9, Alternating, 12]][q, t]
Out[15]=	-6 -4 1 1 1 3 1 2 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 24 9 22 8 20 8 20 7 18 7 18 6 q t q t q t q t q t q t 3 4 2 3 5 2 2 ------ + ------ + ------ + ------ + ------ + ------ + ------ + 16 6 16 5 14 5 14 4 12 4 12 3 10 3 q t q t q t q t q t q t q t 2 2 2 ------ + ----- + ---- 10 2 8 2 6 q t q t q t