L8a7

L8a6

L8a8

	Visit L8a7's page at Knotilus! Visit L8a7's page at the original Knot Atlas!
	L8a7 is [math]\displaystyle{ 8^2_{14} }[/math] in the Rolfsen table of links.

L8a7 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X10,5,11,6 X12,3,13,4 X16,11,5,12 X2,9,3,10 X8,13,9,14
Gauss code	{1, -7, 5, -3}, {4, -1, 2, -8, 7, -4, 6, -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(2)^3+4 t(1) t(2)^2-4 t(2)^2-4 t(1) t(2)+4 t(2)+t(1)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{5}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{4}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{7}{q^{11/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z-a^9 z^{-1} -a^7 z^3+2 a^7 z+3 a^7 z^{-1} -3 a^5 z^3-5 a^5 z-2 a^5 z^{-1} -a^3 z^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-8 a^{10} z^4+5 a^{10} z^2+a^{10}+2 a^9 z^7-a^9 z^5-5 a^9 z^3+2 a^9 z-a^9 z^{-1} +8 a^8 z^6-16 a^8 z^4+4 a^8 z^2+3 a^8+2 a^7 z^7+4 a^7 z^5-12 a^7 z^3+6 a^7 z-3 a^7 z^{-1} +5 a^6 z^6-5 a^6 z^4-a^6 z^2+3 a^6+6 a^5 z^5-8 a^5 z^3+5 a^5 z-2 a^5 z^{-1} +3 a^4 z^4+a^3 z^3 }[/math] (db)

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ \frac{19}{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:L8a7/V 2,1 Data:L8a7/V 3,1 Data:L8a7/V 4,1 Data:L8a7/V 4,2 Data:L8a7/V 4,3 Data:L8a7/V 5,1 Data:L8a7/V 5,2 Data:L8a7/V 5,3 Data:L8a7/V 5,4 Data:L8a7/V 6,1 Data:L8a7/V 6,2 Data:L8a7/V 6,3 Data:L8a7/V 6,4 Data:L8a7/V 6,5 Data:L8a7/V 6,6 Data:L8a7/V 6,7 Data:L8a7/V 6,8 Data:L8a7/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 L8a7. 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               3 1 -2 -6             3     3 -8           2 3     1 -10         5 3       2 -12       2 3         1 -14     2 4           -2 -16   1 2             1 -18   2               -2 -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{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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, 7]]
Out[2]=	8
In[3]:=	PD[Link[8, Alternating, 7]]
Out[3]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[10, 5, 11, 6], X[12, 3, 13, 4], X[16, 11, 5, 12], X[2, 9, 3, 10], X[8, 13, 9, 14]]
In[4]:=	GaussCode[Link[8, Alternating, 7]]
Out[4]=	GaussCode[{1, -7, 5, -3}, {4, -1, 2, -8, 7, -4, 6, -5, 8, -2, 3, -6}]
In[5]:=	BR[Link[8, Alternating, 7]]
Out[5]=	BR[Link[8, Alternating, 7]]
In[6]:=	alex = Alexander[Link[8, Alternating, 7]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, Alternating, 7]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, Alternating, 7]], KnotSignature[Link[8, Alternating, 7]]}
Out[9]=	{Infinity, -3}
In[10]:=	J=Jones[Link[8, Alternating, 7]][q]
Out[10]=	-(19/2) 3 4 6 7 5 6 3 -(3/2) -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - q 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q 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, Alternating, 7]][q]
Out[12]=	-30 -28 3 -22 -20 3 2 4 -12 2 -8 q - q - --- - q + q + --- + --- + --- + q + --- + q - 26 18 16 14 10 q q q q q 2 -4 -- + q 6 q
In[13]:=	Kauffman[Link[8, Alternating, 7]][a, z]
Out[13]=	5 7 9 6 8 10 2 a 3 a a 5 7 9 -3 a - 3 a - a + ---- + ---- + -- - 5 a z - 6 a z - 2 a z - z z z 11 6 2 8 2 10 2 3 3 5 3 7 3 a z + a z - 4 a z - 5 a z - a z + 8 a z + 12 a z + 9 3 11 3 4 4 6 4 8 4 10 4 5 a z + 2 a z - 3 a z + 5 a z + 16 a z + 8 a z - 5 5 7 5 9 5 11 5 6 6 8 6 10 6 6 a z - 4 a z + a z - a z - 5 a z - 8 a z - 3 a z - 7 7 9 7 2 a z - 2 a z
In[14]:=	{Vassiliev[2][Link[8, Alternating, 7]], Vassiliev[3][Link[8, Alternating, 7]]}
Out[14]=	19 {0, --} 24
In[15]:=	Kh[Link[8, Alternating, 7]][q, t]
Out[15]=	-4 -2 1 2 1 2 2 4 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 20 8 18 7 16 7 16 6 14 6 14 5 q t q t q t q t q t q t 2 3 5 3 2 3 3 3 ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- 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