L8a11

L8a10

L8a12

	Visit L8a11's page at Knotilus! Visit L8a11's page at the original Knot Atlas!
	L8a11 is [math]\displaystyle{ 8^2_{3} }[/math] in the Rolfsen table of links.

L8a11 Further Notes and Views

Knot presentations

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X16,11,7,12 X12,15,13,16 X6718 X4,13,5,14
Gauss code	{1, -2, 3, -8, 4, -7}, {7, -1, 2, -3, 5, -6, 8, -4, 6, -5}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ -\frac{2 t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+3 t(2) t(1)-t(1)-t(2)+2}{t(1) t(2)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{4}{q^{15/2}}-\frac{3}{q^{17/2}}+\frac{2}{q^{19/2}}-\frac{1}{q^{21/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^9+2 z a^9-z^5 a^7-3 z^3 a^7-z a^7+a^7 z^{-1} -z^5 a^5-4 z^3 a^5-4 z a^5-a^5 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^3 a^{13}+z a^{13}-2 z^4 a^{12}+2 z^2 a^{12}-2 z^5 a^{11}+z^3 a^{11}-2 z^6 a^{10}+3 z^4 a^{10}-3 z^2 a^{10}-z^7 a^9+z^5 a^9-2 z^3 a^9+2 z a^9-3 z^6 a^8+7 z^4 a^8-4 z^2 a^8-z^7 a^7+2 z^5 a^7-z a^7+a^7 z^{-1} -z^6 a^6+2 z^4 a^6+z^2 a^6-a^6-z^5 a^5+4 z^3 a^5-4 z a^5+a^5 z^{-1} }[/math] (db)

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ -\frac{193}{16} }[/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:L8a11/V 2,1 Data:L8a11/V 3,1 Data:L8a11/V 4,1 Data:L8a11/V 4,2 Data:L8a11/V 4,3 Data:L8a11/V 5,1 Data:L8a11/V 5,2 Data:L8a11/V 5,3 Data:L8a11/V 5,4 Data:L8a11/V 6,1 Data:L8a11/V 6,2 Data:L8a11/V 6,3 Data:L8a11/V 6,4 Data:L8a11/V 6,5 Data:L8a11/V 6,6 Data:L8a11/V 6,7 Data:L8a11/V 6,8 Data:L8a11/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]-5 is the signature of L8a11. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                 1 1 -6               1 1 0 -8             2     2 -10           1 1     0 -12         3 2       1 -14       1 1         0 -16     2 3           -1 -18   1 2             1 -20   1               -1 -22 1                 1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/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}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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} }[/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, 11]]
Out[2]=	8
In[3]:=	PD[Link[8, Alternating, 11]]
Out[3]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[14, 5, 15, 6], X[16, 11, 7, 12], X[12, 15, 13, 16], X[6, 7, 1, 8], X[4, 13, 5, 14]]
In[4]:=	GaussCode[Link[8, Alternating, 11]]
Out[4]=	GaussCode[{1, -2, 3, -8, 4, -7}, {7, -1, 2, -3, 5, -6, 8, -4, 6, -5}]
In[5]:=	BR[Link[8, Alternating, 11]]
Out[5]=	BR[Link[8, Alternating, 11]]
In[6]:=	alex = Alexander[Link[8, Alternating, 11]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, Alternating, 11]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, Alternating, 11]], KnotSignature[Link[8, Alternating, 11]]}
Out[9]=	{Infinity, -5}
In[10]:=	J=Jones[Link[8, Alternating, 11]][q]
Out[10]=	-(21/2) 2 3 4 4 3 3 -(7/2) -q + ----- - ----- + ----- - ----- + ----- - ---- + q - 19/2 17/2 15/2 13/2 11/2 9/2 q 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, 11]][q]
Out[12]=	-32 -24 -22 2 2 -14 2 -8 q - q + q + --- + --- + q + --- + q 18 16 12 q q q
In[13]:=	Kauffman[Link[8, Alternating, 11]][a, z]
Out[13]=	5 7 6 a a 5 7 9 13 6 2 8 2 -a + -- + -- - 4 a z - a z + 2 a z + a z + a z - 4 a z - z z 10 2 12 2 5 3 9 3 11 3 13 3 6 4 3 a z + 2 a z + 4 a z - 2 a z + a z - a z + 2 a z + 8 4 10 4 12 4 5 5 7 5 9 5 11 5 7 a z + 3 a z - 2 a z - a z + 2 a z + a z - 2 a z - 6 6 8 6 10 6 7 7 9 7 a z - 3 a z - 2 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[8, Alternating, 11]], Vassiliev[3][Link[8, Alternating, 11]]}
Out[14]=	193 {0, -(---)} 16
In[15]:=	Kh[Link[8, Alternating, 11]][q, t]
Out[15]=	-6 -4 1 1 1 2 2 3 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 22 8 20 7 18 7 18 6 16 6 16 5 q t q t q t q t q t q t 1 1 3 2 1 1 2 1 ------ + ------ + ------ + ------ + ------ + ------ + ----- + ---- 14 5 14 4 12 4 12 3 10 3 10 2 8 2 6 q t q t q t q t q t q t q t q t