L8n4

L8n3

L8n5

	Visit L8n4's page at Knotilus! Visit L8n4's page at the original Knot Atlas!
	L8n4 is [math]\displaystyle{ 8^3_{8} }[/math] in the Rolfsen table of links.

L8n4 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X9,16,10,11 X11,10,12,5 X15,1,16,4
Gauss code	{1, -4, -3, 8}, {-2, -1, 5, 3, -6, 7}, {-7, 2, 4, -5, -8, 6}

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ \frac{26}{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:L8n4/V 2,1 Data:L8n4/V 3,1 Data:L8n4/V 4,1 Data:L8n4/V 4,2 Data:L8n4/V 4,3 Data:L8n4/V 5,1 Data:L8n4/V 5,2 Data:L8n4/V 5,3 Data:L8n4/V 5,4 Data:L8n4/V 6,1 Data:L8n4/V 6,2 Data:L8n4/V 6,3 Data:L8n4/V 6,4 Data:L8n4/V 6,5 Data:L8n4/V 6,6 Data:L8n4/V 6,7 Data:L8n4/V 6,8 Data:L8n4/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]-2 is the signature of L8n4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 χ -1             2 2 -3           1 2 1 -5         2     2 -7       1 2     1 -9     1 1       0 -11     1         1 -13 1 1           0 -15 1             1

Integral Khovanov Homology (db, data source)

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