L9a47

L9a46

L9a48

	Visit L9a47's page at Knotilus! Visit L9a47's page at the original Knot Atlas!
	L9a47 is $9_{2}^{3}$ in the Rolfsen table of links.

L9a47 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(3)^{2}t(2)^{2}+t(1)t(2)^{2}-t(1)t(3)t(2)^{2}+2t(3)t(2)^{2}-t(2)^{2}-t(1)t(3)^{2}t(2)+2t(3)^{2}t(2)-2t(1)t(2)+4t(1)t(3)t(2)-4t(3)t(2)+t(2)+t(1)t(3)^{2}-t(3)^{2}+t(1)-2t(1)t(3)+t(3)}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{-8}+3q^{-7}-5q^{-6}+8q^{-5}-8q^{-4}+10q^{-3}-7q^{-2}+q+6q^{-1}-3$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{8}+3z^{2}a^{6}+a^{6}z^{-2}+3a^{6}-2z^{4}a^{4}-4z^{2}a^{4}-2a^{4}z^{-2}-5a^{4}-z^{4}a^{2}+z^{2}a^{2}+a^{2}z^{-2}+3a^{2}+z^{2}$ (db)
Kauffman polynomial	$z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-7z^{4}a^{8}+5z^{2}a^{8}-2a^{8}+3z^{7}a^{7}-3z^{5}a^{7}-4z^{3}a^{7}+3za^{7}+z^{8}a^{6}+8z^{6}a^{6}-24z^{4}a^{6}+21z^{2}a^{6}+a^{6}z^{-2}-8a^{6}+7z^{7}a^{5}-10z^{5}a^{5}+5za^{5}-2a^{5}z^{-1}+z^{8}a^{4}+10z^{6}a^{4}-26z^{4}a^{4}+24z^{2}a^{4}+2a^{4}z^{-2}-9a^{4}+4z^{7}a^{3}-3z^{5}a^{3}-z^{3}a^{3}+3za^{3}-2a^{3}z^{-1}+5z^{6}a^{2}-8z^{4}a^{2}+7z^{2}a^{2}+a^{2}z^{-2}-4a^{2}+3z^{5}a-3z^{3}a+z^{4}-z^{2}$ (db)

Vassiliev invariants

V2 and V3:

(0, ${\frac {11}{3}}$)

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:L9a47/V 2,1 Data:L9a47/V 3,1 Data:L9a47/V 4,1 Data:L9a47/V 4,2 Data:L9a47/V 4,3 Data:L9a47/V 5,1 Data:L9a47/V 5,2 Data:L9a47/V 5,3 Data:L9a47/V 5,4 Data:L9a47/V 6,1 Data:L9a47/V 6,2 Data:L9a47/V 6,3 Data:L9a47/V 6,4 Data:L9a47/V 6,5 Data:L9a47/V 6,6 Data:L9a47/V 6,7 Data:L9a47/V 6,8 Data:L9a47/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 L9a47. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                   1 1 1                 2   -2 -1               4 1   3 -3             4 3     -1 -5           6 3       3 -7         4 6         2 -9       4 4           0 -11     2 5             3 -13   1 3               -2 -15   2                 2 -17 1                   -1

Integral Khovanov Homology (db, data source)

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

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, 47]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 47]]
Out[3]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[10, 13, 5, 14], X[18, 15, 11, 16], X[14, 7, 15, 8], X[8, 18, 9, 17], X[16, 10, 17, 9], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[4]:=	GaussCode[Link[9, Alternating, 47]]
Out[4]=	GaussCode[{1, -8, 2, -9}, {8, -1, 5, -6, 7, -3}, {9, -2, 3, -5, 4, -7, 6, -4}]
In[5]:=	BR[Link[9, Alternating, 47]]
Out[5]=	BR[Link[9, Alternating, 47]]
In[6]:=	alex = Alexander[Link[9, Alternating, 47]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 47]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 47]], KnotSignature[Link[9, Alternating, 47]]}
Out[9]=	{Infinity, -2}
In[10]:=	J=Jones[Link[9, Alternating, 47]][q]
Out[10]=	-8 3 5 8 8 10 7 6 -3 - q + -- - -- + -- - -- + -- - -- + - + q 7 6 5 4 3 2 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, 47]][q]
Out[12]=	-26 -24 2 2 5 2 5 3 4 4 3 -1 - q - q + --- + --- + --- + --- + --- + --- + -- + -- + -- - 22 18 16 14 12 10 8 6 2 q q q q q q q q q 2 4 q + q
In[13]:=	Kauffman[Link[9, Alternating, 47]][a, z]
Out[13]=	2 4 6 3 5 2 4 6 8 a 2 a a 2 a 2 a 3 -4 a - 9 a - 8 a - 2 a + -- + ---- + -- - ---- - ---- + 3 a z + 2 2 2 z z z z z 5 7 9 2 2 2 4 2 6 2 5 a z + 3 a z + a z - z + 7 a z + 24 a z + 21 a z + 8 2 3 3 3 7 3 9 3 4 2 4 5 a z - 3 a z - a z - 4 a z - 2 a z + z - 8 a z - 4 4 6 4 8 4 5 3 5 5 5 26 a z - 24 a z - 7 a z + 3 a z - 3 a z - 10 a z - 7 5 9 5 2 6 4 6 6 6 8 6 3 7 3 a z + a z + 5 a z + 10 a z + 8 a z + 3 a z + 4 a z + 5 7 7 7 4 8 6 8 7 a z + 3 a z + a z + a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 47]], Vassiliev[3][Link[9, Alternating, 47]]}
Out[14]=	11 {0, --} 3
In[15]:=	Kh[Link[9, Alternating, 47]][q, t]
Out[15]=	3 4 1 2 1 3 2 5 4 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 q q t q t q t q t q t q t q t 4 4 6 6 3 4 t 3 2 ----- + ----- + ----- + ----- + ---- + ---- + - + 2 q t + q t 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t