L9a21

L9a20

L9a22

	Visit L9a21's page at Knotilus! Visit L9a21's page at the original Knot Atlas!
	L9a21 is $9_{34}^{2}$ in the Rolfsen table of links.

L9a21 Further Notes and Views

Knot presentations

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,8,15,7 X18,14,7,13 X6,17,1,18 X16,11,17,12 X12,6,13,5 X4,16,5,15
Gauss code	{1, -2, 3, -9, 8, -6}, {4, -1, 2, -3, 7, -8, 5, -4, 9, -7, 6, -5}

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, ${\frac {31}{48}}$)

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:L9a21/V 2,1 Data:L9a21/V 3,1 Data:L9a21/V 4,1 Data:L9a21/V 4,2 Data:L9a21/V 4,3 Data:L9a21/V 5,1 Data:L9a21/V 5,2 Data:L9a21/V 5,3 Data:L9a21/V 5,4 Data:L9a21/V 6,1 Data:L9a21/V 6,2 Data:L9a21/V 6,3 Data:L9a21/V 6,4 Data:L9a21/V 6,5 Data:L9a21/V 6,6 Data:L9a21/V 6,7 Data:L9a21/V 6,8 Data:L9a21/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 L9a21. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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, 21]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 21]]
Out[3]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[14, 8, 15, 7], X[18, 14, 7, 13], X[6, 17, 1, 18], X[16, 11, 17, 12], X[12, 6, 13, 5], X[4, 16, 5, 15]]
In[4]:=	GaussCode[Link[9, Alternating, 21]]
Out[4]=	GaussCode[{1, -2, 3, -9, 8, -6}, {4, -1, 2, -3, 7, -8, 5, -4, 9, -7, 6, -5}]
In[5]:=	BR[Link[9, Alternating, 21]]
Out[5]=	BR[Link[9, Alternating, 21]]
In[6]:=	alex = Alexander[Link[9, Alternating, 21]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 21]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 21]], KnotSignature[Link[9, Alternating, 21]]}
Out[9]=	{Infinity, -1}
In[10]:=	J=Jones[Link[9, Alternating, 21]][q]
Out[10]=	-(11/2) 3 5 8 8 9 3/2 q - ---- + ---- - ---- + ---- - ------- + 7 Sqrt[q] - 5 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 3 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 21]][q]
Out[12]=	-16 -14 -12 2 3 -6 4 -2 2 4 6 2 - q + q - q + --- + -- + q + -- - q - q - q + q - 10 8 4 q q q 8 10 q + q
In[13]:=	Kauffman[Link[9, Alternating, 21]][a, z]
Out[13]=	3 2 2 a a 3 z 5 2 3 z 2 2 6 2 -a + - + -- + --- + 4 a z - a z - 4 z - ---- - 2 a z + a z + z z a 2 a 3 3 4 2 z 8 z 3 3 3 5 3 4 7 z 2 4 ---- - ---- - 17 a z - 3 a z + 4 a z + 6 z + ---- + 4 a z + 3 a 2 a a 5 5 6 4 4 6 4 z 10 z 5 3 5 5 5 3 z 4 a z - a z - -- + ----- + 19 a z + 5 a z - 3 a z - ---- - 3 a 2 a a 7 2 6 4 6 4 z 7 3 7 8 2 8 a z - 4 a z - ---- - 8 a z - 4 a z - 2 z - 2 a z a
In[14]:=	{Vassiliev[2][Link[9, Alternating, 21]], Vassiliev[3][Link[9, Alternating, 21]]}
Out[14]=	31 {0, --} 48
In[15]:=	Kh[Link[9, Alternating, 21]][q, t]
Out[15]=	5 1 2 1 3 2 5 4 5 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 q q t q t q t q t q t q t q t 4 4 2 2 2 4 2 4 3 6 3 ---- + ---- + 3 t + 4 q t + 2 q t + 3 q t + q t + 2 q t + 4 2 q t q t 8 4 q t