L8n6

L8n5

L8n7

	Visit L8n6's page at Knotilus! Visit L8n6's page at the original Knot Atlas!
	L8n6 is [math]\displaystyle{ 8^3_{10} }[/math] in the Rolfsen table of links.

L8n6 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X10,3,11,4 X11,16,12,13 X7,14,8,15 X13,8,14,9 X15,12,16,5 X2536 X4,9,1,10
Gauss code	{1, -7, 2, -8}, {-5, 4, -6, 3}, {7, -1, -4, 5, 8, -2, -3, 6}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ \frac{(t(3)-1) (t(3)+1) (t(1) t(2)+t(3))}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-2} + q^{-6} + q^{-7} + q^{-9} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{10} z^{-2} -2 a^8 z^{-2} -2 a^8+a^6 z^{-2} +z^4 a^4+4 z^2 a^4+2 a^4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^6-6 a^{10} z^4+10 a^{10} z^2+a^{10} z^{-2} -6 a^{10}+a^9 z^5-6 a^9 z^3+8 a^9 z-2 a^9 z^{-1} +a^8 z^6-7 a^8 z^4+14 a^8 z^2+2 a^8 z^{-2} -9 a^8+a^7 z^5-6 a^7 z^3+8 a^7 z-2 a^7 z^{-1} +a^6 z^{-2} -2 a^6+a^4 z^4-4 a^4 z^2+2 a^4 }[/math] (db)

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ \frac{106}{3} }[/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:L8n6/V 2,1 Data:L8n6/V 3,1 Data:L8n6/V 4,1 Data:L8n6/V 4,2 Data:L8n6/V 4,3 Data:L8n6/V 5,1 Data:L8n6/V 5,2 Data:L8n6/V 5,3 Data:L8n6/V 5,4 Data:L8n6/V 6,1 Data:L8n6/V 6,2 Data:L8n6/V 6,3 Data:L8n6/V 6,4 Data:L8n6/V 6,5 Data:L8n6/V 6,6 Data:L8n6/V 6,7 Data:L8n6/V 6,8 Data:L8n6/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 L8n6. 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 -3                 1 1 -5                 1 1 -7           1 1     0 -9                   0 -11       1 3 1       1 -13         2         2 -15     1             1 -17 1                 1 -19 1                 1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/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, NonAlternating, 6]]
Out[2]=	8
In[3]:=	PD[Link[8, NonAlternating, 6]]
Out[3]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 16, 12, 13], X[7, 14, 8, 15], X[13, 8, 14, 9], X[15, 12, 16, 5], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[4]:=	GaussCode[Link[8, NonAlternating, 6]]
Out[4]=	GaussCode[{1, -7, 2, -8}, {-5, 4, -6, 3}, {7, -1, -4, 5, 8, -2, -3, 6}]
In[5]:=	BR[Link[8, NonAlternating, 6]]
Out[5]=	BR[Link[8, NonAlternating, 6]]
In[6]:=	alex = Alexander[Link[8, NonAlternating, 6]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, NonAlternating, 6]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, NonAlternating, 6]], KnotSignature[Link[8, NonAlternating, 6]]}
Out[9]=	{Infinity, -3}
In[10]:=	J=Jones[Link[8, NonAlternating, 6]][q]
Out[10]=	-9 -7 -6 -2 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, 6]][q]
Out[12]=	-30 3 4 5 4 3 2 -16 -14 -10 -8 q + --- + --- + --- + --- + --- + --- + q + q + q + q + 28 26 24 22 20 18 q q q q q q -6 q
In[13]:=	Kauffman[Link[8, NonAlternating, 6]][a, z]
Out[13]=	6 8 10 7 9 4 6 8 10 a 2 a a 2 a 2 a 7 2 a - 2 a - 9 a - 6 a + -- + ---- + --- - ---- - ---- + 8 a z + 2 2 2 z z z z z 9 4 2 8 2 10 2 7 3 9 3 4 4 8 a z - 4 a z + 14 a z + 10 a z - 6 a z - 6 a z + a z - 8 4 10 4 7 5 9 5 8 6 10 6 7 a z - 6 a z + a z + a z + a z + a z
In[14]:=	{Vassiliev[2][Link[8, NonAlternating, 6]], Vassiliev[3][Link[8, NonAlternating, 6]]}
Out[14]=	106 {0, ---} 3
In[15]:=	Kh[Link[8, NonAlternating, 6]][q, t]
Out[15]=	-5 -3 1 1 1 1 2 3 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 19 8 17 8 15 6 11 5 13 4 11 4 q t q t q t q t q t q t 1 1 1 ------ + ----- + ----- 11 3 7 3 7 2 q t q t q t