L9a41

L9a40

L9a42

	Visit L9a41's page at Knotilus! Visit L9a41's page at the original Knot Atlas!
	L9a41 is $9_{23}^{2}$ in the Rolfsen table of links.

L9a41 Further Notes and Views

Knot presentations

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X18,5,9,6 X14,8,15,7 X16,14,17,13 X8,16,1,15 X6,9,7,10 X4,17,5,18
Gauss code	{1, -2, 3, -9, 4, -8, 5, -7}, {8, -1, 2, -3, 6, -5, 7, -6, 9, -4}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)t(2)+1)(t(2)t(1)-t(1)+1)(t(1)t(2)-t(2)+1)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{3/2}-2{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {6}{q^{9/2}}}-{\frac {4}{q^{11/2}}}+{\frac {2}{q^{13/2}}}-{\frac {1}{q^{15/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{5}z^{5}+4a^{5}z^{3}+5a^{5}z+a^{5}z^{-1}-a^{3}z^{7}-6a^{3}z^{5}-13a^{3}z^{3}-11a^{3}z-a^{3}z^{-1}+az^{5}+4az^{3}+4az$ (db)
Kauffman polynomial	$-z^{3}a^{9}+za^{9}-2z^{4}a^{8}+z^{2}a^{8}-3z^{5}a^{7}+2z^{3}a^{7}-za^{7}-4z^{6}a^{6}+7z^{4}a^{6}-6z^{2}a^{6}-3z^{7}a^{5}+5z^{5}a^{5}-4z^{3}a^{5}+4za^{5}-a^{5}z^{-1}-z^{8}a^{4}-3z^{6}a^{4}+13z^{4}a^{4}-10z^{2}a^{4}+a^{4}-5z^{7}a^{3}+16z^{5}a^{3}-17z^{3}a^{3}+11za^{3}-a^{3}z^{-1}-z^{8}a^{2}+8z^{4}a^{2}-7z^{2}a^{2}-2z^{7}a+8z^{5}a-10z^{3}a+5za-z^{6}+4z^{4}-4z^{2}$ (db)

Vassiliev invariants

V2 and V3:

(0, $-{\frac {125}{24}}$)

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:L9a41/V 2,1 Data:L9a41/V 3,1 Data:L9a41/V 4,1 Data:L9a41/V 4,2 Data:L9a41/V 4,3 Data:L9a41/V 5,1 Data:L9a41/V 5,2 Data:L9a41/V 5,3 Data:L9a41/V 5,4 Data:L9a41/V 6,1 Data:L9a41/V 6,2 Data:L9a41/V 6,3 Data:L9a41/V 6,4 Data:L9a41/V 6,5 Data:L9a41/V 6,6 Data:L9a41/V 6,7 Data:L9a41/V 6,8 Data:L9a41/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 L9a41. 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 4                   1 -1 2                 1   1 0               2 1   -1 -2             4 1     3 -4           2 3       1 -6         4 3         1 -8       2 2           0 -10     2 4             -2 -12   1 3               2 -14   1                 -1 -16 1                   1

Integral Khovanov Homology (db, data source)

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

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, 41]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 41]]
Out[3]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[18, 5, 9, 6], X[14, 8, 15, 7], X[16, 14, 17, 13], X[8, 16, 1, 15], X[6, 9, 7, 10], X[4, 17, 5, 18]]
In[4]:=	GaussCode[Link[9, Alternating, 41]]
Out[4]=	GaussCode[{1, -2, 3, -9, 4, -8, 5, -7}, {8, -1, 2, -3, 6, -5, 7, -6, 9, -4}]
In[5]:=	BR[Link[9, Alternating, 41]]
Out[5]=	BR[Link[9, Alternating, 41]]
In[6]:=	alex = Alexander[Link[9, Alternating, 41]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 41]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 41]], KnotSignature[Link[9, Alternating, 41]]}
Out[9]=	{Infinity, -3}
In[10]:=	J=Jones[Link[9, Alternating, 41]][q]
Out[10]=	-(15/2) 2 4 6 6 5 6 3 -q + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 2 Sqrt[q] + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 41]][q]
Out[12]=	-22 2 -14 -10 4 2 3 -2 4 -1 + q + --- - q - q + -- + -- + -- + q - q 18 8 6 4 q q q q
In[13]:=	Kauffman[Link[9, Alternating, 41]][a, z]
Out[13]=	3 5 4 a a 3 5 7 9 2 a - -- - -- + 5 a z + 11 a z + 4 a z - a z + a z - 4 z - z z 2 2 4 2 6 2 8 2 3 3 3 5 3 7 a z - 10 a z - 6 a z + a z - 10 a z - 17 a z - 4 a z + 7 3 9 3 4 2 4 4 4 6 4 8 4 2 a z - a z + 4 z + 8 a z + 13 a z + 7 a z - 2 a z + 5 3 5 5 5 7 5 6 4 6 6 6 8 a z + 16 a z + 5 a z - 3 a z - z - 3 a z - 4 a z - 7 3 7 5 7 2 8 4 8 2 a z - 5 a z - 3 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 41]], Vassiliev[3][Link[9, Alternating, 41]]}
Out[14]=	125 {0, -(---)} 24
In[15]:=	Kh[Link[9, Alternating, 41]][q, t]
Out[15]=	3 4 1 1 1 3 2 4 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 4 2 16 6 14 5 12 5 12 4 10 4 10 3 8 3 q q q t q t q t q t q t q t q t 2 4 3 2 t 2 2 2 4 3 ----- + ----- + ---- + ---- + 2 t + -- + t + q t + q t 8 2 6 2 6 4 2 q t q t q t q t q