L8n2

L8n1

L8n3

	Visit L8n2's page at Knotilus! Visit L8n2's page at the original Knot Atlas!
	L8n2 is [math]\displaystyle{ 8^2_{15} }[/math] in the Rolfsen table of links.

L8n2 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X14,7,15,8 X15,1,16,4 X9,12,10,13 X3849 X5,11,6,10 X11,5,12,16 X2,14,3,13
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-1) (v-1)}{\sqrt{u} \sqrt{v}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{5/2}+q^{3/2}-2 \sqrt{q}+\frac{1}{\sqrt{q}}-\frac{1}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{1}{q^{7/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^3+a^3 z^{-1} -z^3 a-3 z a-2 a z^{-1} +2 z a^{-1} +2 a^{-1} z^{-1} - a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^2 z^6-z^6-a^3 z^5-2 a z^5-z^5 a^{-1} +4 a^2 z^4+4 z^4+4 a^3 z^3+8 a z^3+4 z^3 a^{-1} -3 a^2 z^2-z^2 a^{-2} -4 z^2-3 a^3 z-8 a z-6 z a^{-1} -z a^{-3} +1+a^3 z^{-1} +2 a z^{-1} +2 a^{-1} z^{-1} + a^{-3} z^{-1} }[/math] (db)

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ -\frac{1}{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:L8n2/V 2,1 Data:L8n2/V 3,1 Data:L8n2/V 4,1 Data:L8n2/V 4,2 Data:L8n2/V 4,3 Data:L8n2/V 5,1 Data:L8n2/V 5,2 Data:L8n2/V 5,3 Data:L8n2/V 5,4 Data:L8n2/V 6,1 Data:L8n2/V 6,2 Data:L8n2/V 6,3 Data:L8n2/V 6,4 Data:L8n2/V 6,5 Data:L8n2/V 6,6 Data:L8n2/V 6,7 Data:L8n2/V 6,8 Data:L8n2/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 L8n2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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