L9a38

L9a37

L9a39

	Visit L9a38's page at Knotilus! Visit L9a38's page at the original Knot Atlas!
	L9a38 is $9_{5}^{2}$ in the Rolfsen table of links.

L9a38 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(1)t(2)+1)^{2}}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-2q^{9/2}+3q^{7/2}-{\frac {1}{q^{7/2}}}-5q^{5/2}+{\frac {2}{q^{5/2}}}+5q^{3/2}-{\frac {3}{q^{3/2}}}+q^{11/2}-6{\sqrt {q}}+{\frac {4}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+az^{5}-6z^{5}a^{-1}+z^{5}a^{-3}+4az^{3}-12z^{3}a^{-1}+4z^{3}a^{-3}+4az-8za^{-1}+4za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}-2z^{2}a^{-6}+2z^{5}a^{-5}-4z^{3}a^{-5}+za^{-5}+2z^{6}a^{-4}-3z^{4}a^{-4}+z^{2}a^{-4}+2z^{7}a^{-3}+a^{3}z^{5}-5z^{5}a^{-3}-3a^{3}z^{3}+8z^{3}a^{-3}+a^{3}z-4za^{-3}+z^{8}a^{-2}+2a^{2}z^{6}-z^{6}a^{-2}-6a^{2}z^{4}+3a^{2}z^{2}+3z^{2}a^{-2}+2az^{7}+4z^{7}a^{-1}-6az^{5}-14z^{5}a^{-1}+6az^{3}+21z^{3}a^{-1}-4az-10za^{-1}+az^{-1}+a^{-1}z^{-1}+z^{8}-z^{6}-2z^{4}+3z^{2}-1$ (db)

Vassiliev invariants

V2 and V3:

(0, -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:L9a38/V 2,1 Data:L9a38/V 3,1 Data:L9a38/V 4,1 Data:L9a38/V 4,2 Data:L9a38/V 4,3 Data:L9a38/V 5,1 Data:L9a38/V 5,2 Data:L9a38/V 5,3 Data:L9a38/V 5,4 Data:L9a38/V 6,1 Data:L9a38/V 6,2 Data:L9a38/V 6,3 Data:L9a38/V 6,4 Data:L9a38/V 6,5 Data:L9a38/V 6,6 Data:L9a38/V 6,7 Data:L9a38/V 6,8 Data:L9a38/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 L9a38. 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                 1   1 8               2 1   -1 6             3 1     2 4           2 2       0 2         4 3         1 0       2 4           2 -2     1 2             -1 -4   1 2               1 -6   1                 -1 -8 1                   1

Integral Khovanov Homology (db, data source)

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

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, 38]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 38]]
Out[3]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[4, 9, 5, 10], X[14, 6, 15, 5], X[18, 14, 9, 13], X[16, 8, 17, 7], X[6, 16, 7, 15], X[8, 18, 1, 17]]
In[4]:=	GaussCode[Link[9, Alternating, 38]]
Out[4]=	GaussCode[{1, -2, 3, -4, 5, -8, 7, -9}, {4, -1, 2, -3, 6, -5, 8, -7, 9, -6}]
In[5]:=	BR[Link[9, Alternating, 38]]
Out[5]=	BR[Link[9, Alternating, 38]]
In[6]:=	alex = Alexander[Link[9, Alternating, 38]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 38]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 38]], KnotSignature[Link[9, Alternating, 38]]}
Out[9]=	{Infinity, 1}
In[10]:=	J=Jones[Link[9, Alternating, 38]][q]
Out[10]=	-(7/2) 2 3 4 3/2 5/2 -q + ---- - ---- + ------- - 6 Sqrt[q] + 5 q - 5 q + 5/2 3/2 Sqrt[q] q q 7/2 9/2 11/2 3 q - 2 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 38]][q]
Out[12]=	-10 -6 4 6 8 10 12 16 2 + q + q + 3 q + q + 2 q + q - q - q
In[13]:=	Kauffman[Link[9, Alternating, 38]][a, z]
Out[13]=	2 2 1 a z 4 z 10 z 3 2 2 z z 1 - --- - - - -- + --- + ---- + 4 a z - a z - 3 z + ---- - -- - a z z 5 3 a 6 4 a a a a 2 3 3 3 4 3 z 2 2 4 z 8 z 21 z 3 3 3 4 z ---- - 3 a z + ---- - ---- - ----- - 6 a z + 3 a z + 2 z - -- + 2 5 3 a 6 a a a a 4 5 5 5 6 3 z 2 4 2 z 5 z 14 z 5 3 5 6 2 z ---- + 6 a z - ---- + ---- + ----- + 6 a z - a z + z - ---- + 4 5 3 a 4 a a a a 6 7 7 8 z 2 6 2 z 4 z 7 8 z -- - 2 a z - ---- - ---- - 2 a z - z - -- 2 3 a 2 a a a
In[14]:=	{Vassiliev[2][Link[9, Alternating, 38]], Vassiliev[3][Link[9, Alternating, 38]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Link[9, Alternating, 38]][q, t]
Out[15]=	2 1 1 1 2 1 2 2 2 4 + 4 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 3 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 12 5 2 q t + 2 q t + 3 q t + q t + 2 q t + q t + q t + q t