L9a15

L9a14

L9a16

	Visit L9a15's page at Knotilus! Visit L9a15's page at the original Knot Atlas!
	L9a15 is $9_{15}^{2}$ in the Rolfsen table of links.

L9a15 Further Notes and Views

Knot presentations

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

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, $-{\frac {15}{2}}$)

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:L9a15/V 2,1 Data:L9a15/V 3,1 Data:L9a15/V 4,1 Data:L9a15/V 4,2 Data:L9a15/V 4,3 Data:L9a15/V 5,1 Data:L9a15/V 5,2 Data:L9a15/V 5,3 Data:L9a15/V 5,4 Data:L9a15/V 6,1 Data:L9a15/V 6,2 Data:L9a15/V 6,3 Data:L9a15/V 6,4 Data:L9a15/V 6,5 Data:L9a15/V 6,6 Data:L9a15/V 6,7 Data:L9a15/V 6,8 Data:L9a15/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 L9a15. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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