L9a43

L9a42

L9a44

	Visit L9a43's page at Knotilus! Visit L9a43's page at the original Knot Atlas!
	L9a43 is $9_{4}^{3}$ in the Rolfsen table of links.

L9a43 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(3)^{3}+t(2)t(3)^{3}-t(3)^{3}-3t(1)t(3)^{2}+2t(1)t(2)t(3)^{2}-3t(2)t(3)^{2}+2t(3)^{2}+3t(1)t(3)-2t(1)t(2)t(3)+3t(2)t(3)-2t(3)-t(1)+t(1)t(2)-t(2)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-11}+2q^{-10}-5q^{-9}+8q^{-8}-8q^{-7}+10q^{-6}-7q^{-5}+7q^{-4}-3q^{-3}+q^{-2}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{12}z^{-2}+4a^{10}z^{-2}+4a^{10}-6a^{8}z^{2}-5a^{8}z^{-2}-11a^{8}+3a^{6}z^{4}+8a^{6}z^{2}+2a^{6}z^{-2}+7a^{6}+a^{4}z^{4}+a^{4}z^{2}$ (db)
Kauffman polynomial	$a^{13}z^{5}-3a^{13}z^{3}+3a^{13}z-a^{13}z^{-1}+2a^{12}z^{6}-4a^{12}z^{4}+3a^{12}z^{2}+a^{12}z^{-2}-2a^{12}+2a^{11}z^{7}+a^{11}z^{5}-11a^{11}z^{3}+13a^{11}z-5a^{11}z^{-1}+a^{10}z^{8}+6a^{10}z^{6}-16a^{10}z^{4}+14a^{10}z^{2}+4a^{10}z^{-2}-10a^{10}+6a^{9}z^{7}-3a^{9}z^{5}-16a^{9}z^{3}+21a^{9}z-9a^{9}z^{-1}+a^{8}z^{8}+10a^{8}z^{6}-24a^{8}z^{4}+23a^{8}z^{2}+5a^{8}z^{-2}-14a^{8}+4a^{7}z^{7}-10a^{7}z^{3}+11a^{7}z-5a^{7}z^{-1}+6a^{6}z^{6}-11a^{6}z^{4}+11a^{6}z^{2}+2a^{6}z^{-2}-7a^{6}+3a^{5}z^{5}-2a^{5}z^{3}+a^{4}z^{4}-a^{4}z^{2}$ (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:L9a43/V 2,1 Data:L9a43/V 3,1 Data:L9a43/V 4,1 Data:L9a43/V 4,2 Data:L9a43/V 4,3 Data:L9a43/V 5,1 Data:L9a43/V 5,2 Data:L9a43/V 5,3 Data:L9a43/V 5,4 Data:L9a43/V 6,1 Data:L9a43/V 6,2 Data:L9a43/V 6,3 Data:L9a43/V 6,4 Data:L9a43/V 6,5 Data:L9a43/V 6,6 Data:L9a43/V 6,7 Data:L9a43/V 6,8 Data:L9a43/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 $s=$-4 is the signature of L9a43. 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 -3                   1 1 -5                 3 1 -2 -7               4     4 -9             3 3     0 -11           7 4       3 -13         4 6         2 -15       4 4           0 -17     1 4             3 -19   1 4               -3 -21   1                 1 -23 1                   -1

Integral Khovanov Homology (db, data source)

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

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, 43]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 43]]
Out[3]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 7, 15, 8], X[8, 13, 5, 14], X[16, 11, 17, 12], X[18, 15, 9, 16], X[12, 17, 13, 18], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[4]:=	GaussCode[Link[9, Alternating, 43]]
Out[4]=	GaussCode[{1, -8, 2, -9}, {8, -1, 3, -4}, {9, -2, 5, -7, 4, -3, 6, -5, 7, -6}]
In[5]:=	BR[Link[9, Alternating, 43]]
Out[5]=	BR[Link[9, Alternating, 43]]
In[6]:=	alex = Alexander[Link[9, Alternating, 43]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 43]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 43]], KnotSignature[Link[9, Alternating, 43]]}
Out[9]=	{Infinity, -4}
In[10]:=	J=Jones[Link[9, Alternating, 43]][q]
Out[10]=	-11 2 5 8 8 10 7 7 3 -2 -q + --- - -- + -- - -- + -- - -- + -- - -- + q 10 9 8 7 6 5 4 3 q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 43]][q]
Out[12]=	-36 3 2 -30 2 4 5 6 8 4 6 -q - --- - --- - q - --- + --- + --- + --- + --- + --- + --- + 34 32 28 26 24 22 20 18 16 q q q q q q q q q -14 -12 2 2 -6 q + q + --- - -- + q 10 8 q q
In[13]:=	Kauffman[Link[9, Alternating, 43]][a, z]
Out[13]=	6 8 10 12 7 6 8 10 12 2 a 5 a 4 a a 5 a -7 a - 14 a - 10 a - 2 a + ---- + ---- + ----- + --- - ---- - 2 2 2 2 z z z z z 9 11 13 9 a 5 a a 7 9 11 13 4 2 ---- - ----- - --- + 11 a z + 21 a z + 13 a z + 3 a z - a z + z z z 6 2 8 2 10 2 12 2 5 3 7 3 11 a z + 23 a z + 14 a z + 3 a z - 2 a z - 10 a z - 9 3 11 3 13 3 4 4 6 4 8 4 16 a z - 11 a z - 3 a z + a z - 11 a z - 24 a z - 10 4 12 4 5 5 9 5 11 5 13 5 16 a z - 4 a z + 3 a z - 3 a z + a z + a z + 6 6 8 6 10 6 12 6 7 7 9 7 6 a z + 10 a z + 6 a z + 2 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, 43]], Vassiliev[3][Link[9, Alternating, 43]]}
Out[14]=	1 {0, -} 2
In[15]:=	Kh[Link[9, Alternating, 43]][q, t]
Out[15]=	-5 -3 1 1 1 4 1 4 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 23 9 21 8 19 8 19 7 17 7 17 6 q t q t q t q t q t q t 4 4 4 6 7 4 3 3 ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 15 6 15 5 13 5 13 4 11 4 11 3 9 3 9 2 q t q t q t q t q t q t q t q t 4 3 ----- + ---- 7 2 5 q t q t