L9a13

L9a12

L9a14

	Visit L9a13's page at Knotilus! Visit L9a13's page at the original Knot Atlas!
	L9a13 is $9_{16}^{2}$ in the Rolfsen table of links.

L9a13 Further Notes and Views

Knot presentations

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

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, ${\frac {199}{24}}$)

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:L9a13/V 2,1 Data:L9a13/V 3,1 Data:L9a13/V 4,1 Data:L9a13/V 4,2 Data:L9a13/V 4,3 Data:L9a13/V 5,1 Data:L9a13/V 5,2 Data:L9a13/V 5,3 Data:L9a13/V 5,4 Data:L9a13/V 6,1 Data:L9a13/V 6,2 Data:L9a13/V 6,3 Data:L9a13/V 6,4 Data:L9a13/V 6,5 Data:L9a13/V 6,6 Data:L9a13/V 6,7 Data:L9a13/V 6,8 Data:L9a13/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 L9a13. 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 -2                   1 1 -4                 3 1 -2 -6               3     3 -8             3 3     0 -10           5 3       2 -12         3 4         1 -14       3 4           -1 -16     1 3             2 -18   1 3               -2 -20   1                 1 -22 1                   -1

Integral Khovanov Homology (db, data source)

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

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, 13]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 13]]
Out[3]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[18, 13, 5, 14], X[14, 7, 15, 8], X[8, 17, 9, 18], X[16, 9, 17, 10], X[10, 15, 11, 16], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[4]:=	GaussCode[Link[9, Alternating, 13]]
Out[4]=	GaussCode[{1, -8, 2, -9}, {8, -1, 4, -5, 6, -7, 9, -2, 3, -4, 7, -6, 5, -3}]
In[5]:=	BR[Link[9, Alternating, 13]]
Out[5]=	BR[Link[9, Alternating, 13]]
In[6]:=	alex = Alexander[Link[9, Alternating, 13]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 13]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 13]], KnotSignature[Link[9, Alternating, 13]]}
Out[9]=	{Infinity, -3}
In[10]:=	J=Jones[Link[9, Alternating, 13]][q]
Out[10]=	-(21/2) 2 4 6 7 8 6 6 3 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q q -(3/2) q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 13]][q]
Out[12]=	-34 2 -28 -26 3 -22 2 2 -16 3 2 -q - --- - q - q + --- + q + --- + --- + q + --- + --- + 32 24 20 18 14 10 q q q q q q -8 2 -4 q - -- + q 6 q
In[13]:=	Kauffman[Link[9, Alternating, 13]][a, z]
Out[13]=	5 9 11 6 8 10 12 a 2 a a 5 9 -a + 3 a + 5 a + 2 a + -- - ---- - --- - 4 a z + 6 a z + z z z 11 6 2 8 2 10 2 12 2 3 3 5 3 2 a z - a z - 13 a z - 17 a z - 5 a z - a z + 7 a z + 7 3 9 3 11 3 4 4 6 4 8 4 3 a z - 12 a z - 7 a z - 3 a z + 7 a z + 21 a z + 10 4 12 4 5 5 7 5 9 5 11 5 15 a z + 4 a z - 6 a z + 3 a z + 16 a z + 7 a z - 6 6 8 6 10 6 12 6 7 7 9 7 6 a z - 7 a z - 2 a z - a z - 4 a z - 6 a z - 11 7 8 8 10 8 2 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 13]], Vassiliev[3][Link[9, Alternating, 13]]}
Out[14]=	199 {0, ---} 24
In[15]:=	Kh[Link[9, Alternating, 13]][q, t]
Out[15]=	-4 -2 1 1 1 3 1 3 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 22 9 20 8 18 8 18 7 16 7 16 6 q t q t q t q t q t q t 3 4 3 4 5 3 3 3 ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 14 6 14 5 12 5 12 4 10 4 10 3 8 3 8 2 q t q t q t q t q t q t q t q t 3 3 ----- + ---- 6 2 4 q t q t