L7a3

L7a2

L7a4

	Visit L7a3's page at Knotilus! Visit L7a3's page at the original Knot Atlas!
	L7a3 is [math]\displaystyle{ 7^2_4 }[/math] in the Rolfsen table of links.

L7a3 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ -\frac{(t(1)-1) (t(2)-1) \left(t(2)^2+1\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 3 q^{9/2}-3 q^{7/2}+2 q^{5/2}-3 q^{3/2}+q^{13/2}-2 q^{11/2}+\sqrt{q}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{-5} +2 z a^{-5} + a^{-5} z^{-1} -z^5 a^{-3} -4 z^3 a^{-3} -5 z a^{-3} -3 a^{-3} z^{-1} +z^3 a^{-1} +3 z a^{-1} +2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^2 a^{-8} +2 z^3 a^{-7} +3 z^4 a^{-6} -3 z^2 a^{-6} + a^{-6} +3 z^5 a^{-5} -6 z^3 a^{-5} +4 z a^{-5} - a^{-5} z^{-1} +z^6 a^{-4} +z^4 a^{-4} -6 z^2 a^{-4} +3 a^{-4} +4 z^5 a^{-3} -12 z^3 a^{-3} +9 z a^{-3} -3 a^{-3} z^{-1} +z^6 a^{-2} -2 z^4 a^{-2} -2 z^2 a^{-2} +3 a^{-2} +z^5 a^{-1} -4 z^3 a^{-1} +5 z a^{-1} -2 a^{-1} z^{-1} }[/math] (db)

Vassiliev invariants

V2 and V3:

(0, -3)

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:L7a3/V 2,1 Data:L7a3/V 3,1 Data:L7a3/V 4,1 Data:L7a3/V 4,2 Data:L7a3/V 4,3 Data:L7a3/V 5,1 Data:L7a3/V 5,2 Data:L7a3/V 5,3 Data:L7a3/V 5,4 Data:L7a3/V 6,1 Data:L7a3/V 6,2 Data:L7a3/V 6,3 Data:L7a3/V 6,4 Data:L7a3/V 6,5 Data:L7a3/V 6,6 Data:L7a3/V 6,7 Data:L7a3/V 6,8 Data:L7a3/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 L7a3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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