L7a5

L7a4

L7a6

	Visit L7a5's page at Knotilus! Visit L7a5's page at the original Knot Atlas!
	L7a5 is [math]\displaystyle{ 7^2_2 }[/math] in the Rolfsen table of links.

L7a5 Further Notes and Views

Knot presentations

Planar diagram presentation	X8192 X10,3,11,4 X12,6,13,5 X14,11,7,12 X4,14,5,13 X2738 X6,9,1,10
Gauss code	{1, -6, 2, -5, 3, -7}, {6, -1, 7, -2, 4, -3, 5, -4}

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ \frac{31}{48} }[/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:L7a5/V 2,1 Data:L7a5/V 3,1 Data:L7a5/V 4,1 Data:L7a5/V 4,2 Data:L7a5/V 4,3 Data:L7a5/V 5,1 Data:L7a5/V 5,2 Data:L7a5/V 5,3 Data:L7a5/V 5,4 Data:L7a5/V 6,1 Data:L7a5/V 6,2 Data:L7a5/V 6,3 Data:L7a5/V 6,4 Data:L7a5/V 6,5 Data:L7a5/V 6,6 Data:L7a5/V 6,7 Data:L7a5/V 6,8 Data:L7a5/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 L7a5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 χ 4               1 1 2             1   -1 0           2 1   1 -2         2 2     0 -4       2 1       1 -6     1 3         2 -8   1 1           0 -10   1             1 -12 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=0 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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}^{2} }[/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}^{2}\oplus{\mathbb Z}_2 }[/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 }[/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, 5]]
Out[2]=	7
In[3]:=	PD[Link[7, Alternating, 5]]
Out[3]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[12, 6, 13, 5], X[14, 11, 7, 12], X[4, 14, 5, 13], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[4]:=	GaussCode[Link[7, Alternating, 5]]
Out[4]=	GaussCode[{1, -6, 2, -5, 3, -7}, {6, -1, 7, -2, 4, -3, 5, -4}]
In[5]:=	BR[Link[7, Alternating, 5]]
Out[5]=	BR[Link[7, Alternating, 5]]
In[6]:=	alex = Alexander[Link[7, Alternating, 5]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[7, Alternating, 5]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[7, Alternating, 5]], KnotSignature[Link[7, Alternating, 5]]}
Out[9]=	{Infinity, -1}
In[10]:=	J=Jones[Link[7, Alternating, 5]][q]
Out[10]=	-(11/2) 2 2 4 3 3 3/2 q - ---- + ---- - ---- + ---- - ------- + 2 Sqrt[q] - q 9/2 7/2 5/2 3/2 Sqrt[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, 5]][q]
Out[12]=	-18 -14 2 2 2 2 2 6 1 - q + q + --- + -- + -- + -- - q + q 10 8 6 4 q q q q
In[13]:=	Kauffman[Link[7, Alternating, 5]][a, z]
Out[13]=	3 2 a a z 3 5 2 2 2 4 2 -a + - + -- + - - 2 a z - 6 a z - 3 a z + 2 z + 2 a z + 2 a z + z z a 3 6 2 z 3 3 3 5 3 4 2 4 6 4 2 a z - -- + 2 a z + 8 a z + 5 a z - 2 z - a z - a z - a 5 3 5 5 5 2 6 4 6 2 a z - 4 a z - 2 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[7, Alternating, 5]], Vassiliev[3][Link[7, Alternating, 5]]}
Out[14]=	31 {0, --} 48
In[15]:=	Kh[Link[7, Alternating, 5]][q, t]
Out[15]=	2 1 1 1 1 1 3 2 2 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 q q t q t q t q t q t q t q t 1 2 2 4 2 ---- + ---- + t + q t + q t 4 2 q t q t