L9a49

L9a48

L9a50

	Visit L9a49's page at Knotilus! Visit L9a49's page at the original Knot Atlas!
	L9a49 is $9_{6}^{3}$ in the Rolfsen table of links.

L9a49 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{2}w-uv^{2}+uvw^{2}-3uvw+2uv-uw^{2}+2uw-2v^{2}w+v^{2}-2vw^{2}+3vw-v+w^{2}-w}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{5}+3q^{4}+q^{-4}-4q^{3}-2q^{-3}+7q^{2}+5q^{-2}-7q-6q^{-1}+8$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{-4}+a^{4}+z^{4}a^{-2}-2a^{2}z^{2}+a^{2}z^{-2}+z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}+z^{4}-z^{2}-2z^{-2}-3$ (db)
Kauffman polynomial	$z^{8}a^{-2}+z^{8}+2az^{7}+5z^{7}a^{-1}+3z^{7}a^{-3}+3a^{2}z^{6}+4z^{6}a^{-2}+3z^{6}a^{-4}+4z^{6}+2a^{3}z^{5}+2az^{5}-7z^{5}a^{-1}-6z^{5}a^{-3}+z^{5}a^{-5}+a^{4}z^{4}-4a^{2}z^{4}-13z^{4}a^{-2}-8z^{4}a^{-4}-10z^{4}-2a^{3}z^{3}-7az^{3}-z^{3}a^{-1}+2z^{3}a^{-3}-2z^{3}a^{-5}-2a^{4}z^{2}+4a^{2}z^{2}+11z^{2}a^{-2}+5z^{2}a^{-4}+12z^{2}+6az+6za^{-1}+a^{4}-3a^{2}-5a^{-2}-8-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (db)

Vassiliev invariants

V2 and V3:

(0, 1)

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:L9a49/V 2,1 Data:L9a49/V 3,1 Data:L9a49/V 4,1 Data:L9a49/V 4,2 Data:L9a49/V 4,3 Data:L9a49/V 5,1 Data:L9a49/V 5,2 Data:L9a49/V 5,3 Data:L9a49/V 5,4 Data:L9a49/V 6,1 Data:L9a49/V 6,2 Data:L9a49/V 6,3 Data:L9a49/V 6,4 Data:L9a49/V 6,5 Data:L9a49/V 6,6 Data:L9a49/V 6,7 Data:L9a49/V 6,8 Data:L9a49/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 L9a49. 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             5 2     3 3           4 4       0 1         4 3         1 -1       3 5           2 -3     2 3             -1 -5     3               3 -7 1 2                 -1 -9 1                   1

Integral Khovanov Homology (db, data source)

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

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, 49]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 49]]
Out[3]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[16, 10, 17, 9], X[14, 8, 15, 7], X[18, 14, 11, 13], X[10, 16, 5, 15], X[8, 18, 9, 17], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[4]:=	GaussCode[Link[9, Alternating, 49]]
Out[4]=	GaussCode[{1, -8, 2, -9}, {8, -1, 4, -7, 3, -6}, {9, -2, 5, -4, 6, -3, 7, -5}]
In[5]:=	BR[Link[9, Alternating, 49]]
Out[5]=	BR[Link[9, Alternating, 49]]
In[6]:=	alex = Alexander[Link[9, Alternating, 49]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 49]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 49]], KnotSignature[Link[9, Alternating, 49]]}
Out[9]=	{Infinity, 0}
In[10]:=	J=Jones[Link[9, Alternating, 49]][q]
Out[10]=	-4 2 5 6 2 3 4 5 8 + q - -- + -- - - - 7 q + 7 q - 4 q + 3 q - q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 49]][q]
Out[12]=	-14 -12 -10 2 2 -4 5 2 4 6 3 + q + q - q + -- + -- + q + -- + 5 q + 3 q + 2 q + 8 6 2 q q q 8 10 12 14 16 3 q - q + q + q - q
In[13]:=	Kauffman[Link[9, Alternating, 49]][a, z]
Out[13]=	2 5 2 4 2 1 a 2 2 a 6 z -8 - -- - 3 a + a + -- + ----- + -- - --- - --- + --- + 6 a z + 2 2 2 2 2 a z z a a z a z z 2 2 3 3 3 2 5 z 11 z 2 2 4 2 2 z 2 z z 12 z + ---- + ----- + 4 a z - 2 a z - ---- + ---- - -- - 4 2 5 3 a a a a a 4 4 5 3 3 3 4 8 z 13 z 2 4 4 4 z 7 a z - 2 a z - 10 z - ---- - ----- - 4 a z + a z + -- - 4 2 5 a a a 5 5 6 6 6 z 7 z 5 3 5 6 3 z 4 z 2 6 ---- - ---- + 2 a z + 2 a z + 4 z + ---- + ---- + 3 a z + 3 a 4 2 a a a 7 7 8 3 z 5 z 7 8 z ---- + ---- + 2 a z + z + -- 3 a 2 a a
In[14]:=	{Vassiliev[2][Link[9, Alternating, 49]], Vassiliev[3][Link[9, Alternating, 49]]}
Out[14]=	{0, 1}
In[15]:=	Kh[Link[9, Alternating, 49]][q, t]
Out[15]=	5 1 1 2 3 2 3 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 4 q t + 4 q t + 5 q t + 2 q t + 2 q t + q t + 2 q t + 11 5 q t