L8n1

L8a21

L8n2

	Visit L8n1's page at Knotilus! Visit L8n1's page at the original Knot Atlas!
	L8n1 is [math]\displaystyle{ 8^2_{16} }[/math] in the Rolfsen table of links.

L8n1 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X9,12,10,13 X3849 X5,11,6,10 X11,5,12,16 X13,2,14,3
Gauss code	{1, 8, -5, -3}, {-6, -1, 2, 5, -4, 6, -7, 4, -8, -2, 3, 7}

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ -\frac{137}{24} }[/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:L8n1/V 2,1 Data:L8n1/V 3,1 Data:L8n1/V 4,1 Data:L8n1/V 4,2 Data:L8n1/V 4,3 Data:L8n1/V 5,1 Data:L8n1/V 5,2 Data:L8n1/V 5,3 Data:L8n1/V 5,4 Data:L8n1/V 6,1 Data:L8n1/V 6,2 Data:L8n1/V 6,3 Data:L8n1/V 6,4 Data:L8n1/V 6,5 Data:L8n1/V 6,6 Data:L8n1/V 6,7 Data:L8n1/V 6,8 Data:L8n1/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]-3 is the signature of L8n1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/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, NonAlternating, 1]]
Out[2]=	8
In[3]:=	PD[Link[8, NonAlternating, 1]]
Out[3]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[9, 12, 10, 13], X[3, 8, 4, 9], X[5, 11, 6, 10], X[11, 5, 12, 16], X[13, 2, 14, 3]]
In[4]:=	GaussCode[Link[8, NonAlternating, 1]]
Out[4]=	GaussCode[{1, 8, -5, -3}, {-6, -1, 2, 5, -4, 6, -7, 4, -8, -2, 3, 7}]
In[5]:=	BR[Link[8, NonAlternating, 1]]
Out[5]=	BR[Link[8, NonAlternating, 1]]
In[6]:=	alex = Alexander[Link[8, NonAlternating, 1]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, NonAlternating, 1]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, NonAlternating, 1]], KnotSignature[Link[8, NonAlternating, 1]]}
Out[9]=	{Infinity, -3}
In[10]:=	J=Jones[Link[8, NonAlternating, 1]][q]
Out[10]=	-2 2 2 2 2 1 ----- + ---- - ---- + ---- - ---- + ------- - Sqrt[q] 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[8, NonAlternating, 1]][q]
Out[12]=	-22 -20 2 -16 -10 -8 -4 -2 2 1 + q + q + --- + q - q + q + q + q + q 18 q
In[13]:=	Kauffman[Link[8, NonAlternating, 1]][a, z]
Out[13]=	3 5 7 4 a 2 a 2 a a 3 5 7 2 2 a + - + ---- + ---- + -- - 4 a z - 8 a z - 7 a z - 3 a z - a z - z z z z 4 2 3 3 3 5 3 2 4 4 4 6 4 a z + 4 a z + 10 a z + 6 a z + 3 a z + 2 a z - a z - 5 3 5 5 5 2 6 4 6 a z - 3 a z - 2 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[8, NonAlternating, 1]], Vassiliev[3][Link[8, NonAlternating, 1]]}
Out[14]=	137 {0, -(---)} 24
In[15]:=	Kh[Link[8, NonAlternating, 1]][q, t]
Out[15]=	-4 2 2 1 1 1 1 1 1 q + -- + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 2 12 4 10 4 10 3 8 3 8 2 6 2 6 q q t q t q t q t q t q t q t 1 t 2 2 ---- + -- + q t 4 2 q t q