L9a29

L9a28

L9a30

	Visit L9a29's page at Knotilus! Visit L9a29's page at the original Knot Atlas!
	L9a29 is $9_{19}^{2}$ in the Rolfsen table of links.

L9a29 Further Notes and Views

Knot presentations

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X6718 X16,11,17,12 X14,6,15,5 X4,16,5,15 X18,13,7,14 X12,17,13,18
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 5, -9, 8, -6, 7, -5, 9, -8}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{4}-u^{2}v^{3}+u^{2}v^{2}-u^{2}v-uv^{4}+uv^{3}-uv^{2}+uv-u-v^{3}+v^{2}-v+1}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {3}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {3}{q^{15/2}}}-{\frac {2}{q^{17/2}}}+{\frac {1}{q^{19/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{5}\right)-4a^{7}z^{3}-4a^{7}z-a^{7}z^{-1}+a^{5}z^{7}+6a^{5}z^{5}+12a^{5}z^{3}+10a^{5}z+3a^{5}z^{-1}-a^{3}z^{5}-5a^{3}z^{3}-7a^{3}z-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{2}+2a^{11}z^{3}+3a^{10}z^{4}-2a^{10}z^{2}+4a^{9}z^{5}-7a^{9}z^{3}+2a^{9}z+4a^{8}z^{6}-10a^{8}z^{4}+5a^{8}z^{2}-a^{8}+3a^{7}z^{7}-9a^{7}z^{5}+6a^{7}z^{3}-3a^{7}z+a^{7}z^{-1}+a^{6}z^{8}-11a^{6}z^{4}+13a^{6}z^{2}-3a^{6}+4a^{5}z^{7}-19a^{5}z^{5}+27a^{5}z^{3}-14a^{5}z+3a^{5}z^{-1}+a^{4}z^{8}-4a^{4}z^{6}+2a^{4}z^{4}+5a^{4}z^{2}-3a^{4}+a^{3}z^{7}-6a^{3}z^{5}+12a^{3}z^{3}-9a^{3}z+2a^{3}z^{-1}$ (db)

Vassiliev invariants

V2 and V3:

(0, ${\frac {247}{48}}$)

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:L9a29/V 2,1 Data:L9a29/V 3,1 Data:L9a29/V 4,1 Data:L9a29/V 4,2 Data:L9a29/V 4,3 Data:L9a29/V 5,1 Data:L9a29/V 5,2 Data:L9a29/V 5,3 Data:L9a29/V 5,4 Data:L9a29/V 6,1 Data:L9a29/V 6,2 Data:L9a29/V 6,3 Data:L9a29/V 6,4 Data:L9a29/V 6,5 Data:L9a29/V 6,6 Data:L9a29/V 6,7 Data:L9a29/V 6,8 Data:L9a29/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 $s=$-5 is the signature of L9a29. 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 0                   1 1 -2                     0 -4               3 1   2 -6             1 1     0 -8           3 2       1 -10         2 2         0 -12       2 2           0 -14     1 2             1 -16   1 2               -1 -18   1                 1 -20 1                   -1

Integral Khovanov Homology (db, data source)

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

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, 29]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 29]]
Out[3]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[6, 7, 1, 8], X[16, 11, 17, 12], X[14, 6, 15, 5], X[4, 16, 5, 15], X[18, 13, 7, 14], X[12, 17, 13, 18]]
In[4]:=	GaussCode[Link[9, Alternating, 29]]
Out[4]=	GaussCode[{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 5, -9, 8, -6, 7, -5, 9, -8}]
In[5]:=	BR[Link[9, Alternating, 29]]
Out[5]=	BR[Link[9, Alternating, 29]]
In[6]:=	alex = Alexander[Link[9, Alternating, 29]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 29]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 29]], KnotSignature[Link[9, Alternating, 29]]}
Out[9]=	{Infinity, -5}
In[10]:=	J=Jones[Link[9, Alternating, 29]][q]
Out[10]=	-(19/2) 2 3 4 4 4 3 3 q - ----- + ----- - ----- + ----- - ---- + ---- - ---- + 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q -(3/2) 1 q - ------- Sqrt[q]
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 29]][q]
Out[12]=	-28 -24 2 2 -10 2 2 -4 -2 -q - q + --- + --- + q + -- + -- + q + q 16 12 8 6 q q q q
In[13]:=	Kauffman[Link[9, Alternating, 29]][a, z]
Out[13]=	3 5 7 4 6 8 2 a 3 a a 3 5 7 3 a + 3 a + a - ---- - ---- - -- + 9 a z + 14 a z + 3 a z - z z z 9 4 2 6 2 8 2 10 2 12 2 2 a z - 5 a z - 13 a z - 5 a z + 2 a z - a z - 3 3 5 3 7 3 9 3 11 3 4 4 12 a z - 27 a z - 6 a z + 7 a z - 2 a z - 2 a z + 6 4 8 4 10 4 3 5 5 5 7 5 11 a z + 10 a z - 3 a z + 6 a z + 19 a z + 9 a z - 9 5 4 6 8 6 3 7 5 7 7 7 4 8 4 a z + 4 a z - 4 a z - a z - 4 a z - 3 a z - a z - 6 8 a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 29]], Vassiliev[3][Link[9, Alternating, 29]]}
Out[14]=	247 {0, ---} 48
In[15]:=	Kh[Link[9, Alternating, 29]][q, t]
Out[15]=	-6 3 1 1 1 2 1 2 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + 4 20 7 18 6 16 6 16 5 14 5 14 4 q q t q t q t q t q t q t 2 2 2 2 3 2 1 t 2 ------ + ------ + ------ + ------ + ----- + ---- + ---- + -- + t 12 4 12 3 10 3 10 2 8 2 8 6 4 q t q t q t q t q t q t q t q