L9a23

L9a22

L9a24

	Visit L9a23's page at Knotilus! Visit L9a23's page at the original Knot Atlas!
	L9a23 is $9_{22}^{2}$ in the Rolfsen table of links.

L9a23 Further Notes and Views

Knot presentations

Planar diagram presentation	X8192 X10,3,11,4 X18,16,7,15 X14,5,15,6 X4,13,5,14 X12,18,13,17 X16,12,17,11 X2738 X6,9,1,10
Gauss code	{1, -8, 2, -5, 4, -9}, {8, -1, 9, -2, 7, -6, 5, -4, 3, -7, 6, -3}

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, $-{\frac {105}{16}}$)

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:L9a23/V 2,1 Data:L9a23/V 3,1 Data:L9a23/V 4,1 Data:L9a23/V 4,2 Data:L9a23/V 4,3 Data:L9a23/V 5,1 Data:L9a23/V 5,2 Data:L9a23/V 5,3 Data:L9a23/V 5,4 Data:L9a23/V 6,1 Data:L9a23/V 6,2 Data:L9a23/V 6,3 Data:L9a23/V 6,4 Data:L9a23/V 6,5 Data:L9a23/V 6,6 Data:L9a23/V 6,7 Data:L9a23/V 6,8 Data:L9a23/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 L9a23. 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                 2   2 2               3 1   -2 0             4 2     2 -2           5 4       -1 -4         3 3         0 -6       3 5           2 -8     2 3             -1 -10     3               3 -12 1 2                 -1 -14 1                   1

Integral Khovanov Homology (db, data source)

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

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, 23]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 23]]
Out[3]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[18, 16, 7, 15], X[14, 5, 15, 6], X[4, 13, 5, 14], X[12, 18, 13, 17], X[16, 12, 17, 11], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[4]:=	GaussCode[Link[9, Alternating, 23]]
Out[4]=	GaussCode[{1, -8, 2, -5, 4, -9}, {8, -1, 9, -2, 7, -6, 5, -4, 3, -7, 6, -3}]
In[5]:=	BR[Link[9, Alternating, 23]]
Out[5]=	BR[Link[9, Alternating, 23]]
In[6]:=	alex = Alexander[Link[9, Alternating, 23]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 23]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 23]], KnotSignature[Link[9, Alternating, 23]]}
Out[9]=	{Infinity, -1}
In[10]:=	J=Jones[Link[9, Alternating, 23]][q]
Out[10]=	-(13/2) 2 5 6 8 8 7 -q + ----- - ---- + ---- - ---- + ---- - ------- + 5 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 3 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 23]][q]
Out[12]=	-20 2 4 -12 3 -6 2 2 6 8 2 + q + --- + --- + q + --- - q - -- - q + q - q 16 14 10 2 q q q q
In[13]:=	Kauffman[Link[9, Alternating, 23]][a, z]
Out[13]=	3 5 2 4 a 3 a 2 a z 3 5 7 1 + 3 a + 3 a - - - ---- - ---- - - + 11 a z + 8 a z - 2 a z - z z z a 2 3 2 z 2 2 4 2 6 2 4 z 3 3 3 2 z + -- - 13 a z - 11 a z - a z + ---- + 2 a z - 15 a z - 2 a a 4 5 3 7 3 4 z 2 4 4 4 6 4 10 a z + 3 a z + 5 z - -- + 15 a z + 13 a z + 4 a z - 2 a 5 3 z 5 3 5 5 5 7 5 6 2 6 4 6 ---- + a z + 13 a z + 8 a z - a z - 4 z - 6 a z - 4 a z - a 6 6 7 3 7 5 7 2 8 4 8 2 a z - 3 a z - 6 a z - 3 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 23]], Vassiliev[3][Link[9, Alternating, 23]]}
Out[14]=	105 {0, -(---)} 16
In[15]:=	Kh[Link[9, Alternating, 23]][q, t]
Out[15]=	4 1 1 2 3 2 3 3 4 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 2 14 6 12 6 12 5 10 4 8 4 8 3 6 3 q q t q t q t q t q t q t q t 5 3 3 5 2 2 2 4 2 6 3 ----- + ----- + ---- + ---- + 2 t + 3 q t + q t + 2 q t + q t 6 2 4 2 4 2 q t q t q t q t