L9a19

L9a18

L9a20

	Visit L9a19's page at Knotilus! Visit L9a19's page at the original Knot Atlas!
	L9a19 is $9_{38}^{2}$ in the Rolfsen table of links.

L9a19 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X2,9,3,10 X12,3,13,4 X10,5,11,6 X18,11,5,12 X4,17,1,18 X16,14,17,13 X14,8,15,7 X8,16,9,15
Gauss code	{1, -2, 3, -6}, {4, -1, 8, -9, 2, -4, 5, -3, 7, -8, 9, -7, 6, -5}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2uv^{3}-5uv^{2}+6uv-2u-2v^{3}+6v^{2}-5v+2}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+7{\sqrt {q}}-{\frac {9}{\sqrt {q}}}+{\frac {10}{q^{3/2}}}-{\frac {11}{q^{5/2}}}+{\frac {8}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-2a^{3}z^{3}-3a^{3}z-a^{3}z^{-1}-az^{5}-az^{3}+z^{3}a^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-2a^{7}z^{3}+a^{7}z+3a^{6}z^{6}-6a^{6}z^{4}+3a^{6}z^{2}+4a^{5}z^{7}-7a^{5}z^{5}+4a^{5}z^{3}-2a^{5}z+a^{5}z^{-1}+2a^{4}z^{8}+4a^{4}z^{6}-13a^{4}z^{4}+7a^{4}z^{2}-a^{4}+10a^{3}z^{7}-18a^{3}z^{5}+10a^{3}z^{3}-3a^{3}z+a^{3}z^{-1}+2a^{2}z^{8}+8a^{2}z^{6}-17a^{2}z^{4}+z^{4}a^{-2}+6a^{2}z^{2}+6az^{7}-6az^{5}+4z^{5}a^{-1}+az^{3}-3z^{3}a^{-1}+7z^{6}-9z^{4}+2z^{2}$ (db)

Vassiliev invariants

V2 and V3:

(0, $-{\frac {77}{24}}$)

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:L9a19/V 2,1 Data:L9a19/V 3,1 Data:L9a19/V 4,1 Data:L9a19/V 4,2 Data:L9a19/V 4,3 Data:L9a19/V 5,1 Data:L9a19/V 5,2 Data:L9a19/V 5,3 Data:L9a19/V 5,4 Data:L9a19/V 6,1 Data:L9a19/V 6,2 Data:L9a19/V 6,3 Data:L9a19/V 6,4 Data:L9a19/V 6,5 Data:L9a19/V 6,6 Data:L9a19/V 6,7 Data:L9a19/V 6,8 Data:L9a19/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 $s=$-1 is the signature of L9a19. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                   1 -1 4                 3   3 2               4 1   -3 0             5 3     2 -2           6 5       -1 -4         5 4         1 -6       3 6           3 -8     3 5             -2 -10   1 4               3 -12   2                 -2 -14 1                   1

Integral Khovanov Homology (db, data source)

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

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, 19]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 19]]
Out[3]=	PD[X[6, 1, 7, 2], X[2, 9, 3, 10], X[12, 3, 13, 4], X[10, 5, 11, 6], X[18, 11, 5, 12], X[4, 17, 1, 18], X[16, 14, 17, 13], X[14, 8, 15, 7], X[8, 16, 9, 15]]
In[4]:=	GaussCode[Link[9, Alternating, 19]]
Out[4]=	GaussCode[{1, -2, 3, -6}, {4, -1, 8, -9, 2, -4, 5, -3, 7, -8, 9, -7, 6, -5}]
In[5]:=	BR[Link[9, Alternating, 19]]
Out[5]=	BR[Link[9, Alternating, 19]]
In[6]:=	alex = Alexander[Link[9, Alternating, 19]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 19]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 19]], KnotSignature[Link[9, Alternating, 19]]}
Out[9]=	{Infinity, -1}
In[10]:=	J=Jones[Link[9, Alternating, 19]][q]
Out[10]=	-(13/2) 3 6 8 11 10 9 -q + ----- - ---- + ---- - ---- + ---- - ------- + 7 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 4 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 19]][q]
Out[12]=	-20 -18 -16 3 -12 3 -8 -6 2 2 2 + q - q + q + --- - q + --- + q + q + -- - -- - 14 10 4 2 q q q q 2 6 8 2 q + 2 q - q
In[13]:=	Kauffman[Link[9, Alternating, 19]][a, z]
Out[13]=	3 5 4 a a 3 5 7 2 2 2 4 2 a - -- - -- + 3 a z + 2 a z - a z - 2 z - 6 a z - 7 a z - z z 3 4 6 2 3 z 3 3 3 5 3 7 3 4 z 3 a z + ---- - a z - 10 a z - 4 a z + 2 a z + 9 z - -- + a 2 a 5 2 4 4 4 6 4 4 z 5 3 5 5 5 17 a z + 13 a z + 6 a z - ---- + 6 a z + 18 a z + 7 a z - a 7 5 6 2 6 4 6 6 6 7 3 7 a z - 7 z - 8 a z - 4 a z - 3 a z - 6 a z - 10 a z - 5 7 2 8 4 8 4 a z - 2 a z - 2 a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 19]], Vassiliev[3][Link[9, Alternating, 19]]}
Out[14]=	77 {0, -(--)} 24
In[15]:=	Kh[Link[9, Alternating, 19]][q, t]
Out[15]=	5 1 2 1 4 3 5 3 5 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 q q t q t q t q t q t q t q t 6 5 4 6 2 2 2 4 2 6 3 ----- + ----- + ---- + ---- + 3 t + 4 q t + q t + 3 q t + q t 6 2 4 2 4 2 q t q t q t q t