L9a24

L9a23

L9a25

	Visit L9a24's page at Knotilus! Visit L9a24's page at the original Knot Atlas!
	L9a24 is $9_{21}^{2}$ in the Rolfsen table of links.

L9a24 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{3}-u^{2}v^{2}+u^{2}v-u^{2}+uv^{4}-3uv^{3}+3uv^{2}-3uv+u-v^{4}+v^{3}-v^{2}+v}{uv^{2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {2}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {6}{q^{9/2}}}-{\frac {6}{q^{11/2}}}+{\frac {4}{q^{13/2}}}-{\frac {2}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{7}-2za^{7}-a^{7}z^{-1}+z^{5}a^{5}+3z^{3}a^{5}+4za^{5}+3a^{5}z^{-1}+z^{5}a^{3}+2z^{3}a^{3}-za^{3}-2a^{3}z^{-1}-z^{3}a-2za$ (db)
Kauffman polynomial	$a^{10}z^{4}-2a^{10}z^{2}+2a^{9}z^{5}-3a^{9}z^{3}+3a^{8}z^{6}-6a^{8}z^{4}+5a^{8}z^{2}-a^{8}+3a^{7}z^{7}-8a^{7}z^{5}+13a^{7}z^{3}-7a^{7}z+a^{7}z^{-1}+a^{6}z^{8}+2a^{6}z^{6}-8a^{6}z^{4}+11a^{6}z^{2}-3a^{6}+5a^{5}z^{7}-15a^{5}z^{5}+22a^{5}z^{3}-14a^{5}z+3a^{5}z^{-1}+a^{4}z^{8}+a^{4}z^{6}-6a^{4}z^{4}+6a^{4}z^{2}-3a^{4}+2a^{3}z^{7}-4a^{3}z^{5}+3a^{3}z^{3}-5a^{3}z+2a^{3}z^{-1}+2a^{2}z^{6}-5a^{2}z^{4}+2a^{2}z^{2}+az^{5}-3az^{3}+2az$ (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:L9a24/V 2,1 Data:L9a24/V 3,1 Data:L9a24/V 4,1 Data:L9a24/V 4,2 Data:L9a24/V 4,3 Data:L9a24/V 5,1 Data:L9a24/V 5,2 Data:L9a24/V 5,3 Data:L9a24/V 5,4 Data:L9a24/V 6,1 Data:L9a24/V 6,2 Data:L9a24/V 6,3 Data:L9a24/V 6,4 Data:L9a24/V 6,5 Data:L9a24/V 6,6 Data:L9a24/V 6,7 Data:L9a24/V 6,8 Data:L9a24/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 $s=$-3 is the signature of L9a24. 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 2                   1 1 0                 1   -1 -2               3 1   2 -4             3 2     -1 -6           4 2       2 -8         3 4         1 -10       3 3           0 -12     1 3             2 -14   1 3               -2 -16   1                 1 -18 1                   -1

Integral Khovanov Homology (db, data source)

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

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, 24]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 24]]
Out[3]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[14, 6, 15, 5], X[16, 11, 17, 12], X[18, 13, 7, 14], X[12, 17, 13, 18], X[4, 16, 5, 15], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[4]:=	GaussCode[Link[9, Alternating, 24]]
Out[4]=	GaussCode[{1, -8, 2, -7, 3, -9}, {8, -1, 9, -2, 4, -6, 5, -3, 7, -4, 6, -5}]
In[5]:=	BR[Link[9, Alternating, 24]]
Out[5]=	BR[Link[9, Alternating, 24]]
In[6]:=	alex = Alexander[Link[9, Alternating, 24]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 24]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 24]], KnotSignature[Link[9, Alternating, 24]]}
Out[9]=	{Infinity, -3}
In[10]:=	J=Jones[Link[9, Alternating, 24]][q]
Out[10]=	-(17/2) 2 4 6 6 7 5 4 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q 2 ------- - Sqrt[q] Sqrt[q]
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 24]][q]
Out[12]=	-26 -22 2 -18 3 3 2 3 -6 -4 2 -q - q - --- + q + --- + --- + --- + -- - q + q + q 20 14 12 10 8 q q q q q
In[13]:=	Kauffman[Link[9, Alternating, 24]][a, z]
Out[13]=	3 5 7 4 6 8 2 a 3 a a 3 5 3 a + 3 a + a - ---- - ---- - -- - 2 a z + 5 a z + 14 a z + z z z 7 2 2 4 2 6 2 8 2 10 2 3 7 a z - 2 a z - 6 a z - 11 a z - 5 a z + 2 a z + 3 a z - 3 3 5 3 7 3 9 3 2 4 4 4 3 a z - 22 a z - 13 a z + 3 a z + 5 a z + 6 a z + 6 4 8 4 10 4 5 3 5 5 5 7 5 8 a z + 6 a z - a z - a z + 4 a z + 15 a z + 8 a z - 9 5 2 6 4 6 6 6 8 6 3 7 5 7 2 a z - 2 a z - a z - 2 a z - 3 a z - 2 a z - 5 a z - 7 7 4 8 6 8 3 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 24]], Vassiliev[3][Link[9, Alternating, 24]]}
Out[14]=	247 {0, ---} 48
In[15]:=	Kh[Link[9, Alternating, 24]][q, t]
Out[15]=	2 3 1 1 1 3 1 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 q q q t q t q t q t q t q t 3 3 3 4 4 2 3 t 2 2 ------ + ------ + ----- + ----- + ----- + ---- + ---- + t + -- + q t 10 4 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q t q