L9a53

L9a52

L9a54

	Visit L9a53's page at Knotilus! Visit L9a53's page at the original Knot Atlas!
	L9a53 is [math]\displaystyle{ 9^3_{12} }[/math] in the Rolfsen table of links.

L9a53 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X18,10,15,9 X8493 X16,5,17,6 X14,17,5,18 X10,16,11,15 X2,12,3,11
Gauss code	{1, -9, 5, -3}, {8, -6, 7, -4}, {6, -1, 2, -5, 4, -8, 9, -2, 3, -7}

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) (t(3)-1)^3}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^4-4 q^3+8 q^2-9 q+12-10 q^{-1} +10 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^4 z^2-a^4+2 a^2 z^4+z^4 a^{-2} +4 a^2 z^2+z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +3 a^2+ a^{-2} -z^6-3 z^4-4 z^2-2 z^{-2} -3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 a^2 z^8+2 z^8+4 a^3 z^7+11 a z^7+7 z^7 a^{-1} +3 a^4 z^6+7 a^2 z^6+8 z^6 a^{-2} +12 z^6+a^5 z^5-5 a^3 z^5-17 a z^5-7 z^5 a^{-1} +4 z^5 a^{-3} -6 a^4 z^4-22 a^2 z^4-11 z^4 a^{-2} +z^4 a^{-4} -28 z^4-2 a^5 z^3-a^3 z^3+3 a z^3-2 z^3 a^{-3} +5 a^4 z^2+17 a^2 z^2+5 z^2 a^{-2} +17 z^2+a^5 z+3 a^3 z+3 a z+z a^{-1} -2 a^4-6 a^2-2 a^{-2} -5-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2} }[/math] (db)

Vassiliev invariants

V2 and V3:

(0, -1)

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:L9a53/V 2,1 Data:L9a53/V 3,1 Data:L9a53/V 4,1 Data:L9a53/V 4,2 Data:L9a53/V 4,3 Data:L9a53/V 5,1 Data:L9a53/V 5,2 Data:L9a53/V 5,3 Data:L9a53/V 5,4 Data:L9a53/V 6,1 Data:L9a53/V 6,2 Data:L9a53/V 6,3 Data:L9a53/V 6,4 Data:L9a53/V 6,5 Data:L9a53/V 6,6 Data:L9a53/V 6,7 Data:L9a53/V 6,8 Data:L9a53/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 L9a53. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                   1 1 7                 3   -3 5               5 1   4 3             4 3     -1 1           8 5       3 -1         6 8         2 -3       4 4           0 -5     2 6             4 -7   1 4               -3 -9   2                 2 -11 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/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[2, Alternating, 1]]
Out[2]=	2
In[3]:=	PD[Link[2, Alternating, 1]]
Out[3]=	PD[X[4, 1, 3, 2], X[2, 3, 1, 4]]
In[4]:=	GaussCode[Link[2, Alternating, 1]]
Out[4]=	GaussCode[{1, -2}, {2, -1}]
In[5]:=	BR[Link[2, Alternating, 1]]
Out[5]=	BR[Link[2, Alternating, 1]]
In[6]:=	alex = Alexander[Link[2, Alternating, 1]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[2, Alternating, 1]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[2, Alternating, 1]], KnotSignature[Link[2, Alternating, 1]]}
Out[9]=	{Infinity, -1}
In[10]:=	J=Jones[Link[2, Alternating, 1]][q]
Out[10]=	-(5/2) 1 -q - ------- Sqrt[q]
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[2, Alternating, 1]][q]
Out[12]=	-10 2 2 2 -2 1 + q + -- + -- + -- + q 8 6 4 q q q
In[13]:=	Kauffman[Link[2, Alternating, 1]][a, z]
Out[13]=	3 2 a a 3 -a + - + -- - a z - a z z z
In[14]:=	{Vassiliev[2][Link[2, Alternating, 1]], Vassiliev[3][Link[2, Alternating, 1]]}
Out[14]=	17 {0, -(--)} 48
In[15]:=	Kh[Link[2, Alternating, 1]][q, t]
Out[15]=	-2 1 1 1 + q + ----- + ----- 6 2 4 2 q t q t