L9a5

L9a4

L9a6

	Visit L9a5's page at Knotilus! Visit L9a5's page at the original Knot Atlas!
	L9a5 is [math]\displaystyle{ 9^2_{30} }[/math] in the Rolfsen table of links.

L9a5 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X18,16,5,15 X16,12,17,11 X12,18,13,17 X8,14,9,13 X2536 X4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, 3, -7, 9, -2, 5, -6, 7, -3, 4, -5, 6, -4}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ \frac{t(1) t(2)^3-3 t(2)^3-4 t(1) t(2)^2+5 t(2)^2+5 t(1) t(2)-4 t(2)-3 t(1)+1}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -3 q^{9/2}+6 q^{7/2}-\frac{1}{q^{7/2}}-8 q^{5/2}+\frac{2}{q^{5/2}}+9 q^{3/2}-\frac{6}{q^{3/2}}+q^{11/2}-9 \sqrt{q}+\frac{7}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^{-1} -2 a z^3+z^3 a^{-1} -2 z^3 a^{-3} +a^3 z-2 a z-z a^{-1} -z a^{-3} +z a^{-5} +a^3 z^{-1} -2 a^{-1} z^{-1} + a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^8 a^{-2} -z^8-3 a z^7-7 z^7 a^{-1} -4 z^7 a^{-3} -2 a^2 z^6-10 z^6 a^{-2} -5 z^6 a^{-4} -7 z^6-a^3 z^5+5 a z^5+11 z^5 a^{-1} +2 z^5 a^{-3} -3 z^5 a^{-5} +3 a^2 z^4+24 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} +19 z^4+3 a^3 z^3-3 a z^3-7 z^3 a^{-1} +2 z^3 a^{-3} +3 z^3 a^{-5} -19 z^2 a^{-2} -5 z^2 a^{-4} +z^2 a^{-6} -13 z^2-3 a^3 z+2 a z+6 z a^{-1} -z a^{-5} -a^2+5 a^{-2} +2 a^{-4} +3+a^3 z^{-1} -2 a^{-1} z^{-1} - a^{-3} z^{-1} }[/math] (db)

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ -\frac{53}{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:L9a5/V 2,1 Data:L9a5/V 3,1 Data:L9a5/V 4,1 Data:L9a5/V 4,2 Data:L9a5/V 4,3 Data:L9a5/V 5,1 Data:L9a5/V 5,2 Data:L9a5/V 5,3 Data:L9a5/V 5,4 Data:L9a5/V 6,1 Data:L9a5/V 6,2 Data:L9a5/V 6,3 Data:L9a5/V 6,4 Data:L9a5/V 6,5 Data:L9a5/V 6,6 Data:L9a5/V 6,7 Data:L9a5/V 6,8 Data:L9a5/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 L9a5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 12                   1 -1 10                 2   2 8               4 1   -3 6             4 2     2 4           5 4       -1 2         4 4         0 0       4 6           2 -2     2 3             -1 -4     4               4 -6 1 2                 -1 -8 1                   1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/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, 5]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 5]]
Out[3]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 8, 15, 7], X[18, 16, 5, 15], X[16, 12, 17, 11], X[12, 18, 13, 17], X[8, 14, 9, 13], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[4]:=	GaussCode[Link[9, Alternating, 5]]
Out[4]=	GaussCode[{1, -8, 2, -9}, {8, -1, 3, -7, 9, -2, 5, -6, 7, -3, 4, -5, 6, -4}]
In[5]:=	BR[Link[9, Alternating, 5]]
Out[5]=	BR[Link[9, Alternating, 5]]
In[6]:=	alex = Alexander[Link[9, Alternating, 5]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 5]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 5]], KnotSignature[Link[9, Alternating, 5]]}
Out[9]=	{Infinity, 1}
In[10]:=	J=Jones[Link[9, Alternating, 5]][q]
Out[10]=	-(7/2) 2 6 7 3/2 5/2 -q + ---- - ---- + ------- - 9 Sqrt[q] + 9 q - 8 q + 5/2 3/2 Sqrt[q] q q 7/2 9/2 11/2 6 q - 3 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 5]][q]
Out[12]=	-12 -10 4 2 2 2 4 6 8 12 14 4 + q + q + -- + -- + -- - q + q - 3 q - q - 2 q + 2 q - 6 4 2 q q q 18 q
In[13]:=	Kauffman[Link[9, Alternating, 5]][a, z]
Out[13]=	3 2 5 2 1 2 a z 6 z 3 3 + -- + -- - a - ---- - --- + -- - -- + --- + 2 a z - 3 a z - 4 2 3 a z z 5 a a a a z a 2 2 2 3 3 3 2 z 5 z 19 z 3 z 2 z 7 z 3 3 3 13 z + -- - ---- - ----- + ---- + ---- - ---- - 3 a z + 3 a z + 6 4 2 5 3 a a a a a a 4 4 4 5 5 5 4 z 7 z 24 z 2 4 3 z 2 z 11 z 5 19 z - -- + ---- + ----- + 3 a z - ---- + ---- + ----- + 5 a z - 6 4 2 5 3 a a a a a a 6 6 7 7 8 3 5 6 5 z 10 z 2 6 4 z 7 z 7 8 z a z - 7 z - ---- - ----- - 2 a z - ---- - ---- - 3 a z - z - -- 4 2 3 a 2 a a a a
In[14]:=	{Vassiliev[2][Link[9, Alternating, 5]], Vassiliev[3][Link[9, Alternating, 5]]}
Out[14]=	53 {0, -(--)} 24
In[15]:=	Kh[Link[9, Alternating, 5]][q, t]
Out[15]=	2 1 1 2 4 2 4 3 2 6 + 4 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 4 q t + 8 4 6 4 6 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 4 2 6 2 6 3 8 3 8 4 10 4 5 q t + 4 q t + 4 q t + 2 q t + 4 q t + q t + 2 q t + 12 5 q t