L7n1

L7a7

L7n2

	Visit L7n1's page at Knotilus! Visit L7n1's page at the original Knot Atlas!
	L7n1 is [math]\displaystyle{ 7^2_7 }[/math] in the Rolfsen table of links.

L7n1 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X5,10,6,11 X3849 X9,14,10,5 X11,2,12,3
Gauss code	{1, 7, -5, -3}, {-4, -1, 2, 5, -6, 4, -7, -2, 3, 6}

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ \frac{31}{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:L7n1/V 2,1 Data:L7n1/V 3,1 Data:L7n1/V 4,1 Data:L7n1/V 4,2 Data:L7n1/V 4,3 Data:L7n1/V 5,1 Data:L7n1/V 5,2 Data:L7n1/V 5,3 Data:L7n1/V 5,4 Data:L7n1/V 6,1 Data:L7n1/V 6,2 Data:L7n1/V 6,3 Data:L7n1/V 6,4 Data:L7n1/V 6,5 Data:L7n1/V 6,6 Data:L7n1/V 6,7 Data:L7n1/V 6,8 Data:L7n1/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]-5 is the signature of L7n1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 χ -4           1 1 -6           1 1 -8       1     1 -10   1         1 -12   2 1       1 -14             0 -16 1           -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/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[7, NonAlternating, 1]]
Out[2]=	7
In[3]:=	PD[Link[7, NonAlternating, 1]]
Out[3]=	PD[X[6, 1, 7, 2], X[12, 7, 13, 8], X[4, 13, 1, 14], X[5, 10, 6, 11], X[3, 8, 4, 9], X[9, 14, 10, 5], X[11, 2, 12, 3]]
In[4]:=	GaussCode[Link[7, NonAlternating, 1]]
Out[4]=	GaussCode[{1, 7, -5, -3}, {-4, -1, 2, 5, -6, 4, -7, -2, 3, 6}]
In[5]:=	BR[Link[7, NonAlternating, 1]]
Out[5]=	BR[Link[7, NonAlternating, 1]]
In[6]:=	alex = Alexander[Link[7, NonAlternating, 1]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[7, NonAlternating, 1]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[7, NonAlternating, 1]], KnotSignature[Link[7, NonAlternating, 1]]}
Out[9]=	{Infinity, -5}
In[10]:=	J=Jones[Link[7, NonAlternating, 1]][q]
Out[10]=	-(15/2) -(13/2) -(9/2) -(5/2) q - q - q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[7, NonAlternating, 1]][q]
Out[12]=	-28 -26 -24 -20 2 3 2 2 -10 -8 -q - q - q + q + --- + --- + --- + --- + q + q 18 16 14 12 q q q q
In[13]:=	Kauffman[Link[7, NonAlternating, 1]][a, z]
Out[13]=	5 7 9 6 8 10 2 a 3 a a 5 7 9 -3 a - 3 a - a + ---- + ---- + -- - 6 a z - 7 a z - a z + z z z 6 2 8 2 5 3 7 3 6 4 8 4 5 5 7 5 4 a z + 4 a z + 5 a z + 5 a z - a z - a z - a z - a z
In[14]:=	{Vassiliev[2][Link[7, NonAlternating, 1]], Vassiliev[3][Link[7, NonAlternating, 1]]}
Out[14]=	31 {0, --} 24
In[15]:=	Kh[Link[7, NonAlternating, 1]][q, t]
Out[15]=	-6 -4 1 2 1 1 1 q + q + ------ + ------ + ------ + ------ + ----- 16 5 12 4 10 4 12 3 8 2 q t q t q t q t q t