L7a7

L7a6

L7n1

	Visit L7a7's page at Knotilus! Visit L7a7's page at the original Knot Atlas!
	L7a7 is [math]\displaystyle{ 7^3_1 }[/math] in the Rolfsen table of links.

L7a7 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X10,3,11,4 X14,12,9,11 X8,14,5,13 X12,8,13,7 X2536 X4,9,1,10
Gauss code	{1, -6, 2, -7}, {6, -1, 5, -4}, {7, -2, 3, -5, 4, -3}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ \frac{u v w+u (-v)-u w+2 u-2 v w+v+w-1}{\sqrt{u} \sqrt{v} \sqrt{w}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-4} -q^3- q^{-3} +3 q^2+4 q^{-2} -3 q-3 q^{-1} +4 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^4 z^{-2} +a^4-2 z^2 a^2-2 a^2 z^{-2} -3 a^2+z^4+2 z^2+ z^{-2} +2-z^2 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^2 z^6+z^6+a^3 z^5+4 a z^5+3 z^5 a^{-1} +a^4 z^4+a^2 z^4+3 z^4 a^{-2} +3 z^4-4 a z^3-3 z^3 a^{-1} +z^3 a^{-3} -3 a^4 z^2-5 a^2 z^2-3 z^2 a^{-2} -5 z^2-3 a^3 z-3 a z+3 a^4+5 a^2+3+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - z^{-2} }[/math] (db)

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ \frac{11}{6} }[/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:L7a7/V 2,1 Data:L7a7/V 3,1 Data:L7a7/V 4,1 Data:L7a7/V 4,2 Data:L7a7/V 4,3 Data:L7a7/V 5,1 Data:L7a7/V 5,2 Data:L7a7/V 5,3 Data:L7a7/V 5,4 Data:L7a7/V 6,1 Data:L7a7/V 6,2 Data:L7a7/V 6,3 Data:L7a7/V 6,4 Data:L7a7/V 6,5 Data:L7a7/V 6,6 Data:L7a7/V 6,7 Data:L7a7/V 6,8 Data:L7a7/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]0 is the signature of L7a7. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 χ 7               1 -1 5             2   2 3           1 1   0 1         3 2     1 -1       3 4       1 -3     1           1 -5     3           3 -7 1 1             0 -9 1               1

Integral Khovanov Homology (db, data source)

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