L9a42

L9a41

L9a43

	Visit L9a42's page at Knotilus! Visit L9a42's page at the original Knot Atlas!
	L9a42 is $9_{41}^{2}$ in the Rolfsen table of links.

L9a42 Further Notes and Views

Knot presentations

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

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, $-{\frac {3}{2}}$)

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:L9a42/V 2,1 Data:L9a42/V 3,1 Data:L9a42/V 4,1 Data:L9a42/V 4,2 Data:L9a42/V 4,3 Data:L9a42/V 5,1 Data:L9a42/V 5,2 Data:L9a42/V 5,3 Data:L9a42/V 5,4 Data:L9a42/V 6,1 Data:L9a42/V 6,2 Data:L9a42/V 6,3 Data:L9a42/V 6,4 Data:L9a42/V 6,5 Data:L9a42/V 6,6 Data:L9a42/V 6,7 Data:L9a42/V 6,8 Data:L9a42/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 L9a42. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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, 42]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 42]]
Out[3]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[18, 5, 9, 6], X[6, 9, 7, 10], X[16, 12, 17, 11], X[14, 8, 15, 7], X[4, 14, 5, 13], X[8, 16, 1, 15], X[2, 17, 3, 18]]
In[4]:=	GaussCode[Link[9, Alternating, 42]]
Out[4]=	GaussCode[{1, -9, 2, -7, 3, -4, 6, -8}, {4, -1, 5, -2, 7, -6, 8, -5, 9, -3}]
In[5]:=	BR[Link[9, Alternating, 42]]
Out[5]=	BR[Link[9, Alternating, 42]]
In[6]:=	alex = Alexander[Link[9, Alternating, 42]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 42]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 42]], KnotSignature[Link[9, Alternating, 42]]}
Out[9]=	{Infinity, 1}
In[10]:=	J=Jones[Link[9, Alternating, 42]][q]
Out[10]=	-(7/2) 3 6 8 3/2 5/2 -q + ---- - ---- + ------- - 10 Sqrt[q] + 9 q - 9 q + 5/2 3/2 Sqrt[q] q q 7/2 9/2 11/2 6 q - 3 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 42]][q]
Out[12]=	-10 -8 2 2 4 8 10 12 14 16 3 + q - q + -- - q + 4 q + 2 q + q - 2 q + q - q 6 q
In[13]:=	Kauffman[Link[9, Alternating, 42]][a, z]
Out[13]=	2 2 2 1 a z 2 z 6 z 3 2 z 3 z 6 z 1 - --- - - - -- + --- + --- + 2 a z - a z - 5 z + -- - ---- - ---- - a z z 5 3 a 6 4 2 a a a a a 3 3 3 4 2 2 3 z 4 z 12 z 3 3 3 4 z 3 a z + ---- - ---- - ----- - 3 a z + 2 a z + 11 z - -- + 5 3 a 6 a a a 4 4 5 5 5 6 z 12 z 2 4 3 z 6 z 17 z 5 3 5 ---- + ----- + 6 a z - ---- + ---- + ----- + 7 a z - a z - 4 2 5 3 a a a a a 6 6 7 7 8 6 5 z 5 z 2 6 5 z 9 z 7 8 2 z 3 z - ---- - ---- - 3 a z - ---- - ---- - 4 a z - 2 z - ---- 4 2 3 a 2 a a a a
In[14]:=	{Vassiliev[2][Link[9, Alternating, 42]], Vassiliev[3][Link[9, Alternating, 42]]}
Out[14]=	3 {0, -(-)} 2
In[15]:=	Kh[Link[9, Alternating, 42]][q, t]
Out[15]=	2 1 2 1 4 2 4 4 2 6 + 6 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 5 q t + 8 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 4 2 6 2 6 3 8 3 8 4 10 4 4 q t + 4 q t + 5 q t + 2 q t + 4 q t + q t + 2 q t + 12 5 q t