L8a9

L8a8

L8a10

	Visit L8a9's page at Knotilus! Visit L8a9's page at the original Knot Atlas!
	L8a9 is [math]\displaystyle{ 8^2_{8} }[/math] in the Rolfsen table of links.

L8a9 Further Notes and Views

Knot presentations

Planar diagram presentation	X8192 X10,4,11,3 X16,10,7,9 X2738 X4,16,5,15 X12,5,13,6 X14,11,15,12 X6,13,1,14
Gauss code	{1, -4, 2, -5, 6, -8}, {4, -1, 3, -2, 7, -6, 8, -7, 5, -3}

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ -\frac{17}{48} }[/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:L8a9/V 2,1 Data:L8a9/V 3,1 Data:L8a9/V 4,1 Data:L8a9/V 4,2 Data:L8a9/V 4,3 Data:L8a9/V 5,1 Data:L8a9/V 5,2 Data:L8a9/V 5,3 Data:L8a9/V 5,4 Data:L8a9/V 6,1 Data:L8a9/V 6,2 Data:L8a9/V 6,3 Data:L8a9/V 6,4 Data:L8a9/V 6,5 Data:L8a9/V 6,6 Data:L8a9/V 6,7 Data:L8a9/V 6,8 Data:L8a9/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 L8a9. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/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}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/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[8, Alternating, 9]]
Out[2]=	8
In[3]:=	PD[Link[8, Alternating, 9]]
Out[3]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[16, 10, 7, 9], X[2, 7, 3, 8], X[4, 16, 5, 15], X[12, 5, 13, 6], X[14, 11, 15, 12], X[6, 13, 1, 14]]
In[4]:=	GaussCode[Link[8, Alternating, 9]]
Out[4]=	GaussCode[{1, -4, 2, -5, 6, -8}, {4, -1, 3, -2, 7, -6, 8, -7, 5, -3}]
In[5]:=	BR[Link[8, Alternating, 9]]
Out[5]=	BR[Link[8, Alternating, 9]]
In[6]:=	alex = Alexander[Link[8, Alternating, 9]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, Alternating, 9]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, Alternating, 9]], KnotSignature[Link[8, Alternating, 9]]}
Out[9]=	{Infinity, -1}
In[10]:=	J=Jones[Link[8, Alternating, 9]][q]
Out[10]=	-(11/2) 3 4 6 6 6 3/2 q - ---- + ---- - ---- + ---- - ------- + 4 Sqrt[q] - 3 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[8, Alternating, 9]][q]
Out[12]=	-18 2 2 -8 2 6 8 3 - q + --- + --- + q + -- + q - q 14 10 4 q q q
In[13]:=	Kauffman[Link[8, Alternating, 9]][a, z]
Out[13]=	3 2 2 a a 2 z 3 5 z 2 2 6 2 -a + - + -- - --- - 7 a z - 7 a z - 2 a z + -- - 2 a z + a z + z z a 2 a 3 4 5 z 3 3 3 5 3 4 z 2 4 4 4 ---- + 13 a z + 13 a z + 5 a z + 3 z - -- + 8 a z + 3 a z - a 2 a 5 6 4 3 z 5 3 5 5 5 6 2 6 a z - ---- - 6 a z - 6 a z - 3 a z - 3 z - 6 a z - a 4 6 7 3 7 3 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[8, Alternating, 9]], Vassiliev[3][Link[8, Alternating, 9]]}
Out[14]=	17 {0, -(--)} 48
In[15]:=	Kh[Link[8, Alternating, 9]][q, t]
Out[15]=	3 1 2 1 2 2 4 3 4 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 q q t q t q t q t q t q t q t 3 3 2 2 2 4 2 6 3 ---- + ---- + 2 t + 2 q t + q t + 2 q t + q t 4 2 q t q t