L7a2

L7a1

L7a3

	Visit L7a2's page at Knotilus! Visit L7a2's page at the original Knot Atlas!
	L7a2 is [math]\displaystyle{ 7^2_5 }[/math] in the Rolfsen table of links.

L7a2 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ \frac{-2 u v^2+2 u v-u-v^3+2 v^2-2 v}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{4}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{17/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^9 z^{-1} +3 a^7 z+3 a^7 z^{-1} -2 a^5 z^3-4 a^5 z-2 a^5 z^{-1} -a^3 z^3-a^3 z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^4-2 a^{10} z^2+a^{10}+2 a^9 z^5-4 a^9 z^3+2 a^9 z-a^9 z^{-1} +a^8 z^6+2 a^8 z^4-7 a^8 z^2+3 a^8+5 a^7 z^5-10 a^7 z^3+8 a^7 z-3 a^7 z^{-1} +a^6 z^6+3 a^6 z^4-6 a^6 z^2+3 a^6+3 a^5 z^5-5 a^5 z^3+5 a^5 z-2 a^5 z^{-1} +2 a^4 z^4-a^4 z^2+a^3 z^3-a^3 z }[/math] (db)

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ \frac{43}{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:L7a2/V 2,1 Data:L7a2/V 3,1 Data:L7a2/V 4,1 Data:L7a2/V 4,2 Data:L7a2/V 4,3 Data:L7a2/V 5,1 Data:L7a2/V 5,2 Data:L7a2/V 5,3 Data:L7a2/V 5,4 Data:L7a2/V 6,1 Data:L7a2/V 6,2 Data:L7a2/V 6,3 Data:L7a2/V 6,4 Data:L7a2/V 6,5 Data:L7a2/V 6,6 Data:L7a2/V 6,7 Data:L7a2/V 6,8 Data:L7a2/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 L7a2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 χ -2               1 1 -4             2 1 -1 -6           2     2 -8         1 2     1 -10       3 2       1 -12     1 2         1 -14   1 2           -1 -16   1             1 -18 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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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, Alternating, 2]]
Out[2]=	7
In[3]:=	PD[Link[7, Alternating, 2]]
Out[3]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 11, 5, 12], X[12, 7, 13, 8], X[8, 13, 9, 14], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[4]:=	GaussCode[Link[7, Alternating, 2]]
Out[4]=	GaussCode[{1, -6, 2, -7}, {6, -1, 4, -5, 7, -2, 3, -4, 5, -3}]
In[5]:=	BR[Link[7, Alternating, 2]]
Out[5]=	BR[Link[7, Alternating, 2]]
In[6]:=	alex = Alexander[Link[7, Alternating, 2]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[7, Alternating, 2]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[7, Alternating, 2]], KnotSignature[Link[7, Alternating, 2]]}
Out[9]=	{Infinity, -3}
In[10]:=	J=Jones[Link[7, Alternating, 2]][q]
Out[10]=	-(17/2) 2 3 4 3 4 2 -(3/2) q - ----- + ----- - ----- + ---- - ---- + ---- - q 15/2 13/2 11/2 9/2 7/2 5/2 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[7, Alternating, 2]][q]
Out[12]=	-28 2 -20 3 2 3 -12 -10 -8 -6 -4 -q - --- + q + --- + --- + --- + q + q + q - q + q 26 18 16 14 q q q q
In[13]:=	Kauffman[Link[7, Alternating, 2]][a, z]
Out[13]=	5 7 9 6 8 10 2 a 3 a a 3 5 7 -3 a - 3 a - a + ---- + ---- + -- + a z - 5 a z - 8 a z - z z z 9 4 2 6 2 8 2 10 2 3 3 5 3 2 a z + a z + 6 a z + 7 a z + 2 a z - a z + 5 a z + 7 3 9 3 4 4 6 4 8 4 10 4 5 5 10 a z + 4 a z - 2 a z - 3 a z - 2 a z - a z - 3 a z - 7 5 9 5 6 6 8 6 5 a z - 2 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[7, Alternating, 2]], Vassiliev[3][Link[7, Alternating, 2]]}
Out[14]=	43 {0, --} 24
In[15]:=	Kh[Link[7, Alternating, 2]][q, t]
Out[15]=	-4 -2 1 1 1 2 1 2 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 18 7 16 6 14 6 14 5 12 5 12 4 q t q t q t q t q t q t 3 2 1 2 2 2 ------ + ------ + ----- + ----- + ----- + ---- 10 4 10 3 8 3 8 2 6 2 4 q t q t q t q t q t q t