L9a4

L9a3

L9a5

	Visit L9a4's page at Knotilus! Visit L9a4's page at the original Knot Atlas!
	L9a4 is [math]\displaystyle{ 9^2_{18} }[/math] in the Rolfsen table of links.

L9a4 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ -\frac{2 (u-1) (v-1) \left(v^2-v+1\right)}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{17/2}-3 q^{15/2}+6 q^{13/2}-7 q^{11/2}+8 q^{9/2}-9 q^{7/2}+6 q^{5/2}-5 q^{3/2}+2 \sqrt{q}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{-7} +z a^{-7} + a^{-7} z^{-1} -z^5 a^{-5} -2 z^3 a^{-5} -2 z a^{-5} -2 a^{-5} z^{-1} -z^5 a^{-3} -2 z^3 a^{-3} -z a^{-3} +z^3 a^{-1} +2 z a^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^8 a^{-4} -z^8 a^{-6} -2 z^7 a^{-3} -6 z^7 a^{-5} -4 z^7 a^{-7} -2 z^6 a^{-2} -4 z^6 a^{-4} -7 z^6 a^{-6} -5 z^6 a^{-8} -z^5 a^{-1} +z^5 a^{-3} +10 z^5 a^{-5} +5 z^5 a^{-7} -3 z^5 a^{-9} +4 z^4 a^{-2} +11 z^4 a^{-4} +16 z^4 a^{-6} +8 z^4 a^{-8} -z^4 a^{-10} +3 z^3 a^{-1} +4 z^3 a^{-3} -6 z^3 a^{-5} -4 z^3 a^{-7} +3 z^3 a^{-9} -z^2 a^{-2} -8 z^2 a^{-4} -13 z^2 a^{-6} -5 z^2 a^{-8} +z^2 a^{-10} -3 z a^{-1} -2 z a^{-3} +3 z a^{-5} +2 z a^{-7} - a^{-2} +3 a^{-4} +5 a^{-6} +2 a^{-8} + a^{-1} z^{-1} -2 a^{-5} z^{-1} - a^{-7} z^{-1} }[/math] (db)

Vassiliev invariants

V2 and V3:

(0, -7)

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:L9a4/V 2,1 Data:L9a4/V 3,1 Data:L9a4/V 4,1 Data:L9a4/V 4,2 Data:L9a4/V 4,3 Data:L9a4/V 5,1 Data:L9a4/V 5,2 Data:L9a4/V 5,3 Data:L9a4/V 5,4 Data:L9a4/V 6,1 Data:L9a4/V 6,2 Data:L9a4/V 6,3 Data:L9a4/V 6,4 Data:L9a4/V 6,5 Data:L9a4/V 6,6 Data:L9a4/V 6,7 Data:L9a4/V 6,8 Data:L9a4/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]3 is the signature of L9a4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 χ 18                   1 -1 16                 2   2 14               4 1   -3 12             3 2     1 10           5 4       -1 8         4 3         1 6       2 5           3 4     3 4             -1 2   1 4               3 0   1                 -1 -2 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=4 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/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}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/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, 4]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 4]]
Out[3]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[16, 10, 17, 9], X[18, 14, 5, 13], X[14, 18, 15, 17], X[8, 16, 9, 15], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[4]:=	GaussCode[Link[9, Alternating, 4]]
Out[4]=	GaussCode[{1, -8, 2, -9}, {8, -1, 3, -7, 4, -2, 9, -3, 5, -6, 7, -4, 6, -5}]
In[5]:=	BR[Link[9, Alternating, 4]]
Out[5]=	BR[Link[9, Alternating, 4]]
In[6]:=	alex = Alexander[Link[9, Alternating, 4]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 4]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 4]], KnotSignature[Link[9, Alternating, 4]]}
Out[9]=	{Infinity, 3}
In[10]:=	J=Jones[Link[9, Alternating, 4]][q]
Out[10]=	1 3/2 5/2 7/2 9/2 11/2 -(-------) + 2 Sqrt[q] - 5 q + 6 q - 9 q + 8 q - 7 q + Sqrt[q] 13/2 15/2 17/2 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, 4]][q]
Out[12]=	-2 2 4 8 10 12 14 16 20 22 q + q + 3 q + 4 q + 2 q + 2 q + 2 q - 2 q - 3 q - q + 24 26 q - q
In[13]:=	Kauffman[Link[9, Alternating, 4]][a, z]
Out[13]=	2 2 5 3 -2 1 2 1 2 z 3 z 2 z 3 z z -- + -- + -- - a - ---- - ---- + --- + --- + --- - --- - --- + --- - 8 6 4 7 5 a z 7 5 3 a 10 a a a a z a z a a a a 2 2 2 2 3 3 3 3 3 4 5 z 13 z 8 z z 3 z 4 z 6 z 4 z 3 z z ---- - ----- - ---- - -- + ---- - ---- - ---- + ---- + ---- - --- + 8 6 4 2 9 7 5 3 a 10 a a a a a a a a a 4 4 4 4 5 5 5 5 5 6 8 z 16 z 11 z 4 z 3 z 5 z 10 z z z 5 z ---- + ----- + ----- + ---- - ---- + ---- + ----- + -- - -- - ---- - 8 6 4 2 9 7 5 3 a 8 a a a a a a a a a 6 6 6 7 7 7 8 8 7 z 4 z 2 z 4 z 6 z 2 z z z ---- - ---- - ---- - ---- - ---- - ---- - -- - -- 6 4 2 7 5 3 6 4 a a a a a a a a
In[14]:=	{Vassiliev[2][Link[9, Alternating, 4]], Vassiliev[3][Link[9, Alternating, 4]]}
Out[14]=	{0, -7}
In[15]:=	Kh[Link[9, Alternating, 4]][q, t]
Out[15]=	2 2 4 1 1 q 4 6 6 2 8 2 4 q + 3 q + ----- + - + -- + 4 q t + 2 q t + 5 q t + 4 q t + 2 2 t t q t 8 3 10 3 10 4 12 4 12 5 14 5 3 q t + 5 q t + 4 q t + 3 q t + 2 q t + 4 q t + 14 6 16 6 18 7 q t + 2 q t + q t