L9a18

L9a17

L9a19

	Visit L9a18's page at Knotilus! Visit L9a18's page at the original Knot Atlas!
	L9a18 is is [math]\displaystyle{ 9^2_{10} }[/math] in the Rolfsen table of links.

L9a18 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X12,4,13,3 X18,8,5,7 X16,10,17,9 X14,12,15,11 X10,16,11,15 X8,18,9,17 X2536 X4,14,1,13
Gauss code	{1, -8, 2, -9}, {8, -1, 3, -7, 4, -6, 5, -2, 9, -5, 6, -4, 7, -3}

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ -\frac{3}{2} }[/math])

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:L9a18/V 2,1 Data:L9a18/V 3,1 Data:L9a18/V 4,1 Data:L9a18/V 4,2 Data:L9a18/V 4,3 Data:L9a18/V 5,1 Data:L9a18/V 5,2 Data:L9a18/V 5,3 Data:L9a18/V 5,4 Data:L9a18/V 6,1 Data:L9a18/V 6,2 Data:L9a18/V 6,3 Data:L9a18/V 6,4 Data:L9a18/V 6,5 Data:L9a18/V 6,6 Data:L9a18/V 6,7 Data:L9a18/V 6,8 Data:L9a18/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 L9a18. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 χ 16                   1 -1 14                 1   1 12               1 1   0 10             2 1     1 8           2 1       -1 6         2 2         0 4       1 2           1 2     2 2             0 0   1 3               2 -2                     0 -4 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=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=7 }[/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, 18]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 18]]
Out[3]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[18, 8, 5, 7], X[16, 10, 17, 9], X[14, 12, 15, 11], X[10, 16, 11, 15], X[8, 18, 9, 17], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[4]:=	GaussCode[Link[9, Alternating, 18]]
Out[4]=	GaussCode[{1, -8, 2, -9}, {8, -1, 3, -7, 4, -6, 5, -2, 9, -5, 6, -4, 7, -3}]
In[5]:=	BR[Link[9, Alternating, 18]]
Out[5]=	BR[Link[9, Alternating, 18]]
In[6]:=	alex = Alexander[Link[9, Alternating, 18]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 18]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 18]], KnotSignature[Link[9, Alternating, 18]]}
Out[9]=	{Infinity, 1}
In[10]:=	J=Jones[Link[9, Alternating, 18]][q]
Out[10]=	-(3/2) 1 3/2 5/2 7/2 9/2 -q + ------- - 3 Sqrt[q] + 3 q - 4 q + 4 q - 3 q + Sqrt[q] 11/2 13/2 15/2 2 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, 18]][q]
Out[12]=	-6 -4 -2 2 4 6 12 16 20 24 3 + q + q + q + q + q + q - q + q + q - q
In[13]:=	Kauffman[Link[9, Alternating, 18]][a, z]
Out[13]=	2 2 3 3 1 a 2 z 2 z 2 z 3 z 3 z 10 z 10 z 1 - --- - - + --- + --- + --- + 2 a z - ---- + ---- - ----- - ----- - a z z 7 5 a 8 4 7 5 a a a a a a 3 4 4 4 5 5 5 5 6 z 3 4 4 z z 4 z 9 z 11 z z z z -- - a z - z + ---- - -- - ---- + ---- + ----- + -- - -- - -- + a 8 6 4 7 5 3 a 8 a a a a a a a 6 6 6 7 7 7 8 8 3 z 3 z z 2 z 3 z z z z ---- + ---- - -- - ---- - ---- - -- - -- - -- 6 4 2 7 5 3 6 4 a a a a a a a a
In[14]:=	{Vassiliev[2][Link[9, Alternating, 18]], Vassiliev[3][Link[9, Alternating, 18]]}
Out[14]=	3 {0, -(-)} 2
In[15]:=	Kh[Link[9, Alternating, 18]][q, t]
Out[15]=	2 1 1 2 4 4 2 6 2 6 3 3 + 2 q + ----- + - + 2 q t + q t + 2 q t + 2 q t + 2 q t + 4 2 t q t 8 3 8 4 10 4 10 5 12 5 12 6 14 6 2 q t + q t + 2 q t + q t + q t + q t + q t + 16 7 q t