L9a44

L9a43

L9a45

	Visit L9a44's page at Knotilus! Visit L9a44's page at the original Knot Atlas!
	L9a44 is $9_{3}^{3}$ in the Rolfsen table of links.

L9a44 Further Notes and Views

Knot presentations

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

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, ${\frac {17}{6}}$)

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:L9a44/V 2,1 Data:L9a44/V 3,1 Data:L9a44/V 4,1 Data:L9a44/V 4,2 Data:L9a44/V 4,3 Data:L9a44/V 5,1 Data:L9a44/V 5,2 Data:L9a44/V 5,3 Data:L9a44/V 5,4 Data:L9a44/V 6,1 Data:L9a44/V 6,2 Data:L9a44/V 6,3 Data:L9a44/V 6,4 Data:L9a44/V 6,5 Data:L9a44/V 6,6 Data:L9a44/V 6,7 Data:L9a44/V 6,8 Data:L9a44/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=$2 is the signature of L9a44. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 13                   1 -1 11                 2   2 9               4 1   -3 7             3 2     1 5           4 4       0 3         4 3         1 1       4 7           3 -1     1 1             0 -3     4               4 -5 1 1                 0 -7 1                   1

Integral Khovanov Homology (db, data source)

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

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, 44]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 44]]
Out[3]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 14, 9, 13], X[16, 12, 17, 11], X[12, 18, 13, 17], X[8, 16, 5, 15], X[14, 8, 15, 7], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[4]:=	GaussCode[Link[9, Alternating, 44]]
Out[4]=	GaussCode[{1, -8, 2, -9}, {8, -1, 7, -6}, {9, -2, 4, -5, 3, -7, 6, -4, 5, -3}]
In[5]:=	BR[Link[9, Alternating, 44]]
Out[5]=	BR[Link[9, Alternating, 44]]
In[6]:=	alex = Alexander[Link[9, Alternating, 44]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 44]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 44]], KnotSignature[Link[9, Alternating, 44]]}
Out[9]=	{Infinity, 2}
In[10]:=	J=Jones[Link[9, Alternating, 44]][q]
Out[10]=	-3 -2 5 2 3 4 5 6 -5 + q - q + - + 8 q - 7 q + 7 q - 6 q + 3 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 44]][q]
Out[12]=	-10 2 3 7 6 2 4 6 8 10 12 7 + q + -- + -- + -- + -- + 6 q + q + 2 q - 3 q - q - 2 q - 8 6 4 2 q q q q 14 16 18 2 q + q - q
In[13]:=	Kauffman[Link[9, Alternating, 44]][a, z]
Out[13]=	2 2 10 2 5 1 4 2 a 1 5 9 -14 - -- - -- - 7 a + -- + ----- + ----- + ---- - ---- - ---- - --- - 4 2 2 4 2 2 2 2 5 3 a z a a z a z a z z a z a z 2 2 5 a 3 z 13 z 21 z 2 6 z 16 z 2 2 --- + --- + ---- + ---- + 11 a z + 19 z + ---- + ----- + 9 a z + z 5 3 a 4 2 a a a a 3 3 3 3 4 4 4 z 6 z 12 z 11 z 3 4 3 z 11 z 21 z -- - ---- - ----- - ----- - 6 a z - 12 z + ---- - ----- - ----- - 7 5 3 a 6 4 2 a a a a a a 5 5 5 6 6 7 2 4 6 z z 8 z 5 6 7 z 7 z 2 6 4 z 5 a z + ---- - -- - ---- - a z + z + ---- + ---- + a z + ---- + 5 3 a 4 2 3 a a a a a 7 8 5 z 7 8 z ---- + a z + z + -- a 2 a
In[14]:=	{Vassiliev[2][Link[9, Alternating, 44]], Vassiliev[3][Link[9, Alternating, 44]]}
Out[14]=	17 {0, --} 6
In[15]:=	Kh[Link[9, Alternating, 44]][q, t]
Out[15]=	3 1 1 1 4 1 1 4 q 7 q + 4 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 7 4 5 4 5 3 3 2 2 q t t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 3 q t + 4 q t + 4 q t + 3 q t + 2 q t + 4 q t + q t + 11 4 13 5 2 q t + q t