L7a1

L6n1

L7a2

	Visit L7a1's page at Knotilus! Visit L7a1's page at the original Knot Atlas!
	L7a1 is [math]\displaystyle{ 7^2_6 }[/math] in the Rolfsen table of links.

L7a1 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X10,6,11,5 X8493 X14,10,5,9 X2,12,3,11
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-1) (v-1) \left(v^2-v+1\right)}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{9/2}+3 q^{7/2}-4 q^{5/2}+\frac{1}{q^{5/2}}+4 q^{3/2}-\frac{3}{q^{3/2}}-5 \sqrt{q}+\frac{3}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^{-3} -z a^{-3} +z^5 a^{-1} -a z^3+3 z^3 a^{-1} -a z+2 z a^{-1} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^3 a^{-5} +3 z^4 a^{-4} -2 z^2 a^{-4} +4 z^5 a^{-3} -5 z^3 a^{-3} +2 z a^{-3} +2 z^6 a^{-2} +a^2 z^4+z^4 a^{-2} -a^2 z^2-3 z^2 a^{-2} +3 a z^5+7 z^5 a^{-1} -6 a z^3-12 z^3 a^{-1} +2 a z+4 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +2 z^6-z^4-2 z^2-1 }[/math] (db)

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ \frac{1}{2} }[/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:L7a1/V 2,1 Data:L7a1/V 3,1 Data:L7a1/V 4,1 Data:L7a1/V 4,2 Data:L7a1/V 4,3 Data:L7a1/V 5,1 Data:L7a1/V 5,2 Data:L7a1/V 5,3 Data:L7a1/V 5,4 Data:L7a1/V 6,1 Data:L7a1/V 6,2 Data:L7a1/V 6,3 Data:L7a1/V 6,4 Data:L7a1/V 6,5 Data:L7a1/V 6,6 Data:L7a1/V 6,7 Data:L7a1/V 6,8 Data:L7a1/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]1 is the signature of L7a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 χ 10               1 1 8             2   -2 6           2 1   1 4         2 2     0 2       3 2       1 0     2 4         2 -2   1 1           0 -4   2             2 -6 1               -1

Integral Khovanov Homology (db, data source)

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