L9a45

L9a44

L9a46

	Visit L9a45's page at Knotilus! Visit L9a45's page at the original Knot Atlas!
	L9a45 is $9_{7}^{3}$ in the Rolfsen table of links.

L9a45 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-2t(2)t(1)+2t(2)t(3)t(1)-2t(3)t(1)+3t(1)+2t(2)-3t(2)t(3)+2t(3)-2}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}}}$ (db)
Jones polynomial	$-q^{5}+3q^{4}-4q^{3}+5q^{2}-6q+7-4q^{-1}+4q^{-2}-q^{-3}+q^{-4}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{-2}+a^{4}-2z^{2}a^{2}-2a^{2}z^{-2}-3a^{2}+z^{4}+z^{2}+z^{-2}+2+z^{4}a^{-2}+z^{2}a^{-2}-z^{2}a^{-4}$ (db)
Kauffman polynomial	$z^{5}a^{-5}-2z^{3}a^{-5}+3z^{6}a^{-4}+a^{4}z^{4}-8z^{4}a^{-4}-3a^{4}z^{2}+3z^{2}a^{-4}-a^{4}z^{-2}+3a^{4}+3z^{7}a^{-3}+a^{3}z^{5}-8z^{5}a^{-3}+5z^{3}a^{-3}-3a^{3}z+2a^{3}z^{-1}+z^{8}a^{-2}+a^{2}z^{6}+z^{6}a^{-2}+2a^{2}z^{4}-5z^{4}a^{-2}-6a^{2}z^{2}+2z^{2}a^{-2}-2a^{2}z^{-2}+5a^{2}+az^{7}+4z^{7}a^{-1}+az^{5}-9z^{5}a^{-1}+7z^{3}a^{-1}-3az+2az^{-1}+z^{8}-z^{6}+4z^{4}-4z^{2}-z^{-2}+3$ (db)

Vassiliev invariants

V2 and V3:

(0, ${\frac {11}{6}}$)

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:L9a45/V 2,1 Data:L9a45/V 3,1 Data:L9a45/V 4,1 Data:L9a45/V 4,2 Data:L9a45/V 4,3 Data:L9a45/V 5,1 Data:L9a45/V 5,2 Data:L9a45/V 5,3 Data:L9a45/V 5,4 Data:L9a45/V 6,1 Data:L9a45/V 6,2 Data:L9a45/V 6,3 Data:L9a45/V 6,4 Data:L9a45/V 6,5 Data:L9a45/V 6,6 Data:L9a45/V 6,7 Data:L9a45/V 6,8 Data:L9a45/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=$0 is the signature of L9a45. 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 11                   1 -1 9                 2   2 7               2 1   -1 5             3 2     1 3           3 2       -1 1         4 3         1 -1       3 6           3 -3     1 1             0 -5     3               3 -7 1 1                 0 -9 1                   1

Integral Khovanov Homology (db, data source)

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

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, 45]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 45]]
Out[3]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 12, 9, 11], X[16, 14, 17, 13], X[8, 16, 5, 15], X[14, 8, 15, 7], X[12, 18, 13, 17], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[4]:=	GaussCode[Link[9, Alternating, 45]]
Out[4]=	GaussCode[{1, -8, 2, -9}, {8, -1, 6, -5}, {9, -2, 3, -7, 4, -6, 5, -4, 7, -3}]
In[5]:=	BR[Link[9, Alternating, 45]]
Out[5]=	BR[Link[9, Alternating, 45]]
In[6]:=	alex = Alexander[Link[9, Alternating, 45]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 45]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 45]], KnotSignature[Link[9, Alternating, 45]]}
Out[9]=	{Infinity, 0}
In[10]:=	J=Jones[Link[9, Alternating, 45]][q]
Out[10]=	-4 -3 4 4 2 3 4 5 7 + q - q + -- - - - 6 q + 5 q - 4 q + 3 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 45]][q]
Out[12]=	-14 2 2 5 5 3 5 2 6 8 10 12 2 + q + --- + --- + -- + -- + -- + -- + 2 q - q + q - q + q + 12 10 8 6 4 2 q q q q q q 14 16 q - q
In[13]:=	Kauffman[Link[9, Alternating, 45]][a, z]
Out[13]=	2 4 3 2 4 -2 2 a a 2 a 2 a 3 3 + 5 a + 3 a - z - ---- - -- + --- + ---- - 3 a z - 3 a z - 2 2 z z z z 2 2 3 3 3 2 3 z 2 z 2 2 4 2 2 z 5 z 7 z 4 4 z + ---- + ---- - 6 a z - 3 a z - ---- + ---- + ---- + 4 z - 4 2 5 3 a a a a a 4 4 5 5 5 8 z 5 z 2 4 4 4 z 8 z 9 z 5 3 5 ---- - ---- + 2 a z + a z + -- - ---- - ---- + a z + a z - 4 2 5 3 a a a a a 6 6 7 7 8 6 3 z z 2 6 3 z 4 z 7 8 z z + ---- + -- + a z + ---- + ---- + a z + z + -- 4 2 3 a 2 a a a a
In[14]:=	{Vassiliev[2][Link[9, Alternating, 45]], Vassiliev[3][Link[9, Alternating, 45]]}
Out[14]=	11 {0, --} 6
In[15]:=	Kh[Link[9, Alternating, 45]][q, t]
Out[15]=	6 1 1 1 3 1 1 3 - + 4 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 3 q t + q 9 4 7 4 7 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 3 q t + 2 q t + 3 q t + 2 q t + 2 q t + q t + 2 q t + 11 5 q t