L9a10

L9a9

L9a11

	Visit L9a10's page at Knotilus! Visit L9a10's page at the original Knot Atlas!
	L9a10 is $9_{36}^{2}$ in the Rolfsen table of links.

L9a10 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X10,4,11,3 X18,12,5,11 X14,9,15,10 X2,14,3,13 X8,15,9,16
Gauss code	{1, -8, 5, -3}, {4, -1, 2, -9, 7, -5, 6, -4, 8, -7, 9, -2, 3, -6}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2(u-1)(v-1)\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-q^{9/2}+{\frac {1}{q^{9/2}}}+3q^{7/2}-{\frac {3}{q^{7/2}}}-6q^{5/2}+{\frac {4}{q^{5/2}}}+7q^{3/2}-{\frac {7}{q^{3/2}}}-8{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$az^{5}+z^{5}a^{-1}-a^{3}z^{3}+2az^{3}+2z^{3}a^{-1}-z^{3}a^{-3}-a^{3}z+2za^{-1}-za^{-3}+a^{3}z^{-1}-2az^{-1}+2a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-5}+a^{4}z^{6}-3a^{4}z^{4}+3z^{4}a^{-4}+2a^{4}z^{2}+3a^{3}z^{7}-11a^{3}z^{5}+6z^{5}a^{-3}+11a^{3}z^{3}-6z^{3}a^{-3}-a^{3}z+3za^{-3}-a^{3}z^{-1}-a^{-3}z^{-1}+2a^{2}z^{8}-3a^{2}z^{6}+7z^{6}a^{-2}-4a^{2}z^{4}-10z^{4}a^{-2}+4a^{2}z^{2}+4z^{2}a^{-2}+8az^{7}+5z^{7}a^{-1}-22az^{5}-5z^{5}a^{-1}+13az^{3}-5z^{3}a^{-1}+2az+6za^{-1}-2az^{-1}-2a^{-1}z^{-1}+2z^{8}+3z^{6}-14z^{4}+6z^{2}-1$ (db)

Vassiliev invariants

V2 and V3:

(0, 1)

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:L9a10/V 2,1 Data:L9a10/V 3,1 Data:L9a10/V 4,1 Data:L9a10/V 4,2 Data:L9a10/V 4,3 Data:L9a10/V 5,1 Data:L9a10/V 5,2 Data:L9a10/V 5,3 Data:L9a10/V 5,4 Data:L9a10/V 6,1 Data:L9a10/V 6,2 Data:L9a10/V 6,3 Data:L9a10/V 6,4 Data:L9a10/V 6,5 Data:L9a10/V 6,6 Data:L9a10/V 6,7 Data:L9a10/V 6,8 Data:L9a10/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 L9a10. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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