L9a20

L9a19

L9a21

	Visit L9a20's page at Knotilus! Visit L9a20's page at the original Knot Atlas!
	L9a20 is [math]\displaystyle{ 9^2_{42} }[/math] in the Rolfsen table of links.

L9a20 Further Notes and Views

Knot presentations

Planar diagram presentation	X8192 X16,11,17,12 X10,4,11,3 X2,15,3,16 X12,5,13,6 X6718 X14,10,15,9 X18,14,7,13 X4,18,5,17
Gauss code	{1, -4, 3, -9, 5, -6}, {6, -1, 7, -3, 2, -5, 8, -7, 4, -2, 9, -8}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ \frac{t(1)^2 t(2)^4-t(1) t(2)^4-3 t(1)^2 t(2)^3+4 t(1) t(2)^3-t(2)^3+3 t(1)^2 t(2)^2-7 t(1) t(2)^2+3 t(2)^2-t(1)^2 t(2)+4 t(1) t(2)-3 t(2)-t(1)+1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{7/2}+4 q^{5/2}-7 q^{3/2}+9 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{11}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^3 z^5-2 a^3 z^3+a^3 z^{-1} +a z^7+4 a z^5-z^5 a^{-1} +4 a z^3-2 z^3 a^{-1} -a z-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -3 a^2 z^8-3 z^8-7 a^3 z^7-13 a z^7-6 z^7 a^{-1} -7 a^4 z^6-6 a^2 z^6-4 z^6 a^{-2} -3 z^6-4 a^5 z^5+8 a^3 z^5+25 a z^5+12 z^5 a^{-1} -z^5 a^{-3} -a^6 z^4+8 a^4 z^4+16 a^2 z^4+7 z^4 a^{-2} +14 z^4+3 a^5 z^3-2 a^3 z^3-12 a z^3-6 z^3 a^{-1} +z^3 a^{-3} -2 a^4 z^2-6 a^2 z^2-2 z^2 a^{-2} -6 z^2-a^3 z-a z-a^2+a^3 z^{-1} +a z^{-1} }[/math] (db)

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ -\frac{17}{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:L9a20/V 2,1 Data:L9a20/V 3,1 Data:L9a20/V 4,1 Data:L9a20/V 4,2 Data:L9a20/V 4,3 Data:L9a20/V 5,1 Data:L9a20/V 5,2 Data:L9a20/V 5,3 Data:L9a20/V 5,4 Data:L9a20/V 6,1 Data:L9a20/V 6,2 Data:L9a20/V 6,3 Data:L9a20/V 6,4 Data:L9a20/V 6,5 Data:L9a20/V 6,6 Data:L9a20/V 6,7 Data:L9a20/V 6,8 Data:L9a20/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 L9a20. 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 8                   1 1 6                 3   -3 4               4 1   3 2             5 3     -2 0           7 4       3 -2         5 6         1 -4       5 6           -1 -6     3 6             3 -8   1 4               -3 -10   3                 3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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[9, Alternating, 20]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 20]]
Out[3]=	PD[X[8, 1, 9, 2], X[16, 11, 17, 12], X[10, 4, 11, 3], X[2, 15, 3, 16], X[12, 5, 13, 6], X[6, 7, 1, 8], X[14, 10, 15, 9], X[18, 14, 7, 13], X[4, 18, 5, 17]]
In[4]:=	GaussCode[Link[9, Alternating, 20]]
Out[4]=	GaussCode[{1, -4, 3, -9, 5, -6}, {6, -1, 7, -3, 2, -5, 8, -7, 4, -2, 9, -8}]
In[5]:=	BR[Link[9, Alternating, 20]]
Out[5]=	BR[Link[9, Alternating, 20]]
In[6]:=	alex = Alexander[Link[9, Alternating, 20]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 20]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 20]], KnotSignature[Link[9, Alternating, 20]]}
Out[9]=	{Infinity, -1}
In[10]:=	J=Jones[Link[9, Alternating, 20]][q]
Out[10]=	-(11/2) 4 7 10 11 12 3/2 q - ---- + ---- - ---- + ---- - ------- + 9 Sqrt[q] - 7 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 4 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 20]][q]
Out[12]=	-16 2 -12 2 2 -6 4 -2 4 6 8 10 4 - q + --- - q + --- + -- - q + -- - q - q + q - 2 q + q 14 10 8 4 q q q q
In[13]:=	Kauffman[Link[9, Alternating, 20]][a, z]
Out[13]=	3 2 3 2 a a 3 2 2 z 2 2 4 2 z -a + - + -- - a z - a z - 6 z - ---- - 6 a z - 2 a z + -- - z z 2 3 a a 3 4 6 z 3 3 3 5 3 4 7 z 2 4 ---- - 12 a z - 2 a z + 3 a z + 14 z + ---- + 16 a z + a 2 a 5 5 4 4 6 4 z 12 z 5 3 5 5 5 6 8 a z - a z - -- + ----- + 25 a z + 8 a z - 4 a z - 3 z - 3 a a 6 7 4 z 2 6 4 6 6 z 7 3 7 8 2 8 ---- - 6 a z - 7 a z - ---- - 13 a z - 7 a z - 3 z - 3 a z 2 a a
In[14]:=	{Vassiliev[2][Link[9, Alternating, 20]], Vassiliev[3][Link[9, Alternating, 20]]}
Out[14]=	17 {0, -(--)} 48
In[15]:=	Kh[Link[9, Alternating, 20]][q, t]
Out[15]=	6 1 3 1 4 3 6 5 7 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 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 6 5 2 2 2 4 2 4 3 6 3 ---- + ---- + 4 t + 5 q t + 3 q t + 4 q t + q t + 3 q t + 4 2 q t q t 8 4 q t