L9a2

L9a1

L9a3

	Visit L9a2's page at Knotilus! Visit L9a2's page at the original Knot Atlas!
	L9a2 is $9_{31}^{2}$ in the Rolfsen table of links.

L9a2 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X12,6,13,5 X8493 X16,10,17,9 X18,12,5,11 X10,18,11,17 X2,14,3,13
Gauss code	{1, -9, 5, -3}, {4, -1, 2, -5, 6, -8, 7, -4, 9, -2, 3, -6, 8, -7}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-t(2)^{3}+t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{15/2}+3q^{13/2}-4q^{11/2}+6q^{9/2}-7q^{7/2}+6q^{5/2}-6q^{3/2}+3{\sqrt {q}}-{\frac {3}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-z^{5}a^{-1}+5z^{5}a^{-3}-z^{5}a^{-5}-3z^{3}a^{-1}+7z^{3}a^{-3}-3z^{3}a^{-5}+za^{-3}-za^{-5}+2a^{-1}z^{-1}-3a^{-3}z^{-1}+a^{-5}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-9}+3z^{4}a^{-8}-2z^{2}a^{-8}+4z^{5}a^{-7}-3z^{3}a^{-7}+4z^{6}a^{-6}-3z^{4}a^{-6}-2z^{2}a^{-6}+a^{-6}+4z^{7}a^{-5}-8z^{5}a^{-5}+6z^{3}a^{-5}-2za^{-5}-a^{-5}z^{-1}+2z^{8}a^{-4}-2z^{6}a^{-4}-3z^{4}a^{-4}+3a^{-4}+7z^{7}a^{-3}-24z^{5}a^{-3}+22z^{3}a^{-3}-3za^{-3}-3a^{-3}z^{-1}+2z^{8}a^{-2}-5z^{6}a^{-2}+z^{2}a^{-2}+3a^{-2}+3z^{7}a^{-1}-12z^{5}a^{-1}+12z^{3}a^{-1}-za^{-1}-2a^{-1}z^{-1}+z^{6}-3z^{4}+z^{2}$ (db)

Vassiliev invariants

V2 and V3:

(0, $-{\frac {3}{2}}$)

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:L9a2/V 2,1 Data:L9a2/V 3,1 Data:L9a2/V 4,1 Data:L9a2/V 4,2 Data:L9a2/V 4,3 Data:L9a2/V 5,1 Data:L9a2/V 5,2 Data:L9a2/V 5,3 Data:L9a2/V 5,4 Data:L9a2/V 6,1 Data:L9a2/V 6,2 Data:L9a2/V 6,3 Data:L9a2/V 6,4 Data:L9a2/V 6,5 Data:L9a2/V 6,6 Data:L9a2/V 6,7 Data:L9a2/V 6,8 Data:L9a2/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 L9a2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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