L9a1

L8n8

L9a2

	Visit L9a1's page at Knotilus! Visit L9a1's page at the original Knot Atlas!
	L9a1 is $9_{32}^{2}$ in the Rolfsen table of links.

L9a1 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}-3t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{13/2}+3q^{11/2}-5q^{9/2}+8q^{7/2}-10q^{5/2}+9q^{3/2}-9{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {1}{q^{5/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-1}+z^{5}a^{-3}-az^{3}+z^{3}a^{-1}+2z^{3}a^{-3}-z^{3}a^{-5}-za^{-1}+2za^{-3}-za^{-5}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{8}a^{-2}-2z^{8}a^{-4}-5z^{7}a^{-1}-9z^{7}a^{-3}-4z^{7}a^{-5}-5z^{6}a^{-2}-2z^{6}a^{-4}-3z^{6}a^{-6}-6z^{6}-4az^{5}+4z^{5}a^{-1}+18z^{5}a^{-3}+9z^{5}a^{-5}-z^{5}a^{-7}-a^{2}z^{4}+11z^{4}a^{-2}+10z^{4}a^{-4}+7z^{4}a^{-6}+7z^{4}+4az^{3}-13z^{3}a^{-3}-7z^{3}a^{-5}+2z^{3}a^{-7}-4z^{2}a^{-2}-7z^{2}a^{-4}-4z^{2}a^{-6}-z^{2}+2za^{-1}+4za^{-3}+2za^{-5}+1-az^{-1}-a^{-1}z^{-1}$ (db)

Vassiliev invariants

V2 and V3:

(0, ${\frac {1}{2}}$)

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:L9a1/V 2,1 Data:L9a1/V 3,1 Data:L9a1/V 4,1 Data:L9a1/V 4,2 Data:L9a1/V 4,3 Data:L9a1/V 5,1 Data:L9a1/V 5,2 Data:L9a1/V 5,3 Data:L9a1/V 5,4 Data:L9a1/V 6,1 Data:L9a1/V 6,2 Data:L9a1/V 6,3 Data:L9a1/V 6,4 Data:L9a1/V 6,5 Data:L9a1/V 6,6 Data:L9a1/V 6,7 Data:L9a1/V 6,8 Data:L9a1/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=$1 is the signature of L9a1. 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 14                   1 1 12                 2   -2 10               3 1   2 8             5 2     -3 6           5 3       2 4         4 5         1 2       5 5           0 0     3 6             3 -2   1 3               -2 -4   3                 3 -6 1                   -1

Integral Khovanov Homology (db, data source)

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

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