L8a13

L8a12

L8a14

	Visit L8a13's page at Knotilus! Visit L8a13's page at the original Knot Atlas!
	L8a13 is [math]\displaystyle{ 8^2_{4} }[/math] in the Rolfsen table of links. Contains two L4a1 configurations.

Symmetric form

(alternate)

National swiss museum

Povray depiction

Knot presentations

Planar diagram presentation	X10,1,11,2 X16,7,9,8 X12,3,13,4 X6,13,7,14 X14,5,15,6 X4,15,5,16 X2,9,3,10 X8,11,1,12
Gauss code	{1, -7, 3, -6, 5, -4, 2, -8}, {7, -1, 8, -3, 4, -5, 6, -2}

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ -\frac{281}{12} }[/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:L8a13/V 2,1 Data:L8a13/V 3,1 Data:L8a13/V 4,1 Data:L8a13/V 4,2 Data:L8a13/V 4,3 Data:L8a13/V 5,1 Data:L8a13/V 5,2 Data:L8a13/V 5,3 Data:L8a13/V 5,4 Data:L8a13/V 6,1 Data:L8a13/V 6,2 Data:L8a13/V 6,3 Data:L8a13/V 6,4 Data:L8a13/V 6,5 Data:L8a13/V 6,6 Data:L8a13/V 6,7 Data:L8a13/V 6,8 Data:L8a13/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 L8a13. 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 -2                 1 1 -4               2 1 -1 -6             2     2 -8           2 2     0 -10         2 2       0 -12       2 2         0 -14     1 2           -1 -16     2             2 -18 1 1               0 -20 1                 1

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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/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} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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, 13]]
Out[2]=	8
In[3]:=	PD[Link[8, Alternating, 13]]
Out[3]=	PD[X[10, 1, 11, 2], X[16, 7, 9, 8], X[12, 3, 13, 4], X[6, 13, 7, 14], X[14, 5, 15, 6], X[4, 15, 5, 16], X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[4]:=	GaussCode[Link[8, Alternating, 13]]
Out[4]=	GaussCode[{1, -7, 3, -6, 5, -4, 2, -8}, {7, -1, 8, -3, 4, -5, 6, -2}]
In[5]:=	BR[Link[8, Alternating, 13]]
Out[5]=	BR[Link[8, Alternating, 13]]
In[6]:=	alex = Alexander[Link[8, Alternating, 13]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, Alternating, 13]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, Alternating, 13]], KnotSignature[Link[8, Alternating, 13]]}
Out[9]=	{Infinity, -3}
In[10]:=	J=Jones[Link[8, Alternating, 13]][q]
Out[10]=	-(19/2) -(17/2) 3 4 4 4 4 2 -q + q - ----- + ----- - ----- + ---- - ---- + ---- - 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q -(3/2) q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[8, Alternating, 13]][q]
Out[12]=	-30 -28 -26 3 -22 -16 -14 -10 -8 -6 -4 q + q + q + --- + q - q + q + q + q - q + q 24 q
In[13]:=	Kauffman[Link[8, Alternating, 13]][a, z]
Out[13]=	7 9 8 a a 3 5 7 9 11 4 2 a - -- - -- + a z - 3 a z + 3 a z + 3 a z - 4 a z + a z - z z 8 2 3 3 5 3 7 3 9 3 11 3 4 4 a z - a z + 4 a z - a z - 2 a z + 4 a z - 2 a z + 6 4 8 4 10 4 5 5 9 5 11 5 6 6 a z + 5 a z + 2 a z - 3 a z + 2 a z - a z - 2 a z - 8 6 10 6 7 7 9 7 3 a z - a z - a z - a z
In[14]:=	{Vassiliev[2][Link[8, Alternating, 13]], Vassiliev[3][Link[8, Alternating, 13]]}
Out[14]=	281 {0, -(---)} 12
In[15]:=	Kh[Link[8, Alternating, 13]][q, t]
Out[15]=	-4 -2 1 1 1 2 1 2 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 20 8 18 8 18 7 16 6 14 6 14 5 q t q t q t q t q t q t 2 2 2 2 2 2 2 2 ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- 12 5 12 4 10 4 10 3 8 3 8 2 6 2 4 q t q t q t q t q t q t q t q t