L7n2

L7n1

L8a1

	Visit L7n2's page at Knotilus! Visit L7n2's page at the original Knot Atlas!
	L7n2 is [math]\displaystyle{ 7^2_8 }[/math] in the Rolfsen table of links.

L7n2 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X12,7,13,8 X13,1,14,4 X5,10,6,11 X3849 X9,14,10,5 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)}{\sqrt{u} \sqrt{v}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{2}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{2}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{9/2}}+\frac{1}{q^{11/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 (-z)-a^5 z^{-1} +a^3 z^3+3 a^3 z+3 a^3 z^{-1} -2 a z-2 a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^4-3 a^6 z^2+a^6+a^5 z^5-3 a^5 z^3+2 a^5 z-a^5 z^{-1} +2 a^4 z^4-5 a^4 z^2+3 a^4+a^3 z^5-3 a^3 z^3+5 a^3 z-3 a^3 z^{-1} +a^2 z^4-2 a^2 z^2+3 a^2+3 a z-2 a z^{-1} }[/math] (db)

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ \frac{5}{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:L7n2/V 2,1 Data:L7n2/V 3,1 Data:L7n2/V 4,1 Data:L7n2/V 4,2 Data:L7n2/V 4,3 Data:L7n2/V 5,1 Data:L7n2/V 5,2 Data:L7n2/V 5,3 Data:L7n2/V 5,4 Data:L7n2/V 6,1 Data:L7n2/V 6,2 Data:L7n2/V 6,3 Data:L7n2/V 6,4 Data:L7n2/V 6,5 Data:L7n2/V 6,6 Data:L7n2/V 6,7 Data:L7n2/V 6,8 Data:L7n2/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 L7n2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 χ 0           2 2 -2         1 2 1 -4       1     1 -6       1     1 -8   1 1       0 -10             0 -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}_2 }[/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}\oplus{\mathbb Z}_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}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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, NonAlternating, 2]]
Out[2]=	7
In[3]:=	PD[Link[7, NonAlternating, 2]]
Out[3]=	PD[X[6, 1, 7, 2], X[12, 7, 13, 8], X[13, 1, 14, 4], X[5, 10, 6, 11], X[3, 8, 4, 9], X[9, 14, 10, 5], X[2, 12, 3, 11]]
In[4]:=	GaussCode[Link[7, NonAlternating, 2]]
Out[4]=	GaussCode[{1, -7, -5, 3}, {-4, -1, 2, 5, -6, 4, 7, -2, -3, 6}]
In[5]:=	BR[Link[7, NonAlternating, 2]]
Out[5]=	BR[Link[7, NonAlternating, 2]]
In[6]:=	alex = Alexander[Link[7, NonAlternating, 2]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[7, NonAlternating, 2]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[7, NonAlternating, 2]], KnotSignature[Link[7, NonAlternating, 2]]}
Out[9]=	{Infinity, -1}
In[10]:=	J=Jones[Link[7, NonAlternating, 2]][q]
Out[10]=	-(11/2) -(9/2) -(7/2) 2 -(3/2) 2 q - q + q - ---- + q - ------- 5/2 Sqrt[q] q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[7, NonAlternating, 2]][q]
Out[12]=	-18 -16 -14 -12 -10 2 3 3 2 2 - q - q - q - q + q + -- + -- + -- + -- 8 6 4 2 q q q q
In[13]:=	Kauffman[Link[7, NonAlternating, 2]][a, z]
Out[13]=	3 5 2 4 6 2 a 3 a a 3 5 -3 a - 3 a - a + --- + ---- + -- - 3 a z - 5 a z - 2 a z + z z z 2 2 4 2 6 2 3 3 5 3 2 4 4 4 2 a z + 5 a z + 3 a z + 3 a z + 3 a z - a z - 2 a z - 6 4 3 5 5 5 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[7, NonAlternating, 2]], Vassiliev[3][Link[7, NonAlternating, 2]]}
Out[14]=	5 {0, -} 2
In[15]:=	Kh[Link[7, NonAlternating, 2]][q, t]
Out[15]=	2 1 1 1 1 1 1 2 + -- + ------ + ----- + ----- + ----- + ----- + ---- 2 12 5 8 4 8 3 6 2 4 2 2 q q t q t q t q t q t q t