L9a10

L9a9

L9a11

	Visit L9a10's page at Knotilus! Visit L9a10's page at the original Knot Atlas!
	L9a10 is [math]\displaystyle{ 9^2_{36} }[/math] 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 [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ \frac{2 (u-1) (v-1) \left(v^2-v+1\right)}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{9/2}+\frac{1}{q^{9/2}}+3 q^{7/2}-\frac{3}{q^{7/2}}-6 q^{5/2}+\frac{4}{q^{5/2}}+7 q^{3/2}-\frac{7}{q^{3/2}}-8 \sqrt{q}+\frac{8}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^5+z^5 a^{-1} -a^3 z^3+2 a z^3+2 z^3 a^{-1} -z^3 a^{-3} -a^3 z+2 z a^{-1} -z a^{-3} +a^3 z^{-1} -2 a z^{-1} +2 a^{-1} z^{-1} - a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^3 a^{-5} +a^4 z^6-3 a^4 z^4+3 z^4 a^{-4} +2 a^4 z^2+3 a^3 z^7-11 a^3 z^5+6 z^5 a^{-3} +11 a^3 z^3-6 z^3 a^{-3} -a^3 z+3 z a^{-3} -a^3 z^{-1} - a^{-3} z^{-1} +2 a^2 z^8-3 a^2 z^6+7 z^6 a^{-2} -4 a^2 z^4-10 z^4 a^{-2} +4 a^2 z^2+4 z^2 a^{-2} +8 a z^7+5 z^7 a^{-1} -22 a z^5-5 z^5 a^{-1} +13 a z^3-5 z^3 a^{-1} +2 a z+6 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +2 z^8+3 z^6-14 z^4+6 z^2-1 }[/math] (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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]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)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math]

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