L9a50

L9a49

L9a51

	Visit L9a50's page at Knotilus! Visit L9a50's page at the original Knot Atlas!
	L9a50 is [math]\displaystyle{ 9^3_{1} }[/math] in the Rolfsen table of links.

L9a50 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ \frac{t(1) t(3)^2 t(2)^2-t(3)^2 t(2)^2-t(1) t(3) t(2)^2+2 t(3) t(2)^2-t(2)^2-2 t(1) t(3)^2 t(2)+t(3)^2 t(2)-t(1) t(2)+2 t(1) t(3) t(2)-2 t(3) t(2)+2 t(2)+t(1) t(3)^2+t(1)-2 t(1) t(3)+t(3)-1}{\sqrt{t(1)} t(2) t(3)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^6+3 q^5-5 q^4+7 q^3+ q^{-3} -7 q^2-2 q^{-2} +8 q+5 q^{-1} -5 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^{-4} -2 z^2 a^{-4} - a^{-4} +z^6 a^{-2} +4 z^4 a^{-2} +a^2 z^2+6 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +2 a^2+4 a^{-2} -2 z^4-6 z^2-2 z^{-2} -5 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^8 a^{-2} +z^8+2 a z^7+6 z^7 a^{-1} +4 z^7 a^{-3} +a^2 z^6+8 z^6 a^{-2} +6 z^6 a^{-4} +3 z^6-6 a z^5-14 z^5 a^{-1} -3 z^5 a^{-3} +5 z^5 a^{-5} -4 a^2 z^4-27 z^4 a^{-2} -9 z^4 a^{-4} +3 z^4 a^{-6} -19 z^4+3 a z^3+3 z^3 a^{-1} -5 z^3 a^{-3} -4 z^3 a^{-5} +z^3 a^{-7} +6 a^2 z^2+23 z^2 a^{-2} +6 z^2 a^{-4} -z^2 a^{-6} +22 z^2+3 a z+5 z a^{-1} +3 z a^{-3} +z a^{-5} -4 a^2-8 a^{-2} -2 a^{-4} -9-2 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:L9a50/V 2,1 Data:L9a50/V 3,1 Data:L9a50/V 4,1 Data:L9a50/V 4,2 Data:L9a50/V 4,3 Data:L9a50/V 5,1 Data:L9a50/V 5,2 Data:L9a50/V 5,3 Data:L9a50/V 5,4 Data:L9a50/V 6,1 Data:L9a50/V 6,2 Data:L9a50/V 6,3 Data:L9a50/V 6,4 Data:L9a50/V 6,5 Data:L9a50/V 6,6 Data:L9a50/V 6,7 Data:L9a50/V 6,8 Data:L9a50/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]2 is the signature of L9a50. 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 13                   1 -1 11                 2   2 9               3 1   -2 7             4 2     2 5           4 4       0 3         4 3         1 1       3 6           3 -1     2 2             0 -3   1 4               3 -5   1                 -1 -7 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=3 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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, 50]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 50]]
Out[3]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[18, 16, 11, 15], X[14, 8, 15, 7], X[10, 12, 5, 11], X[8, 17, 9, 18], X[16, 9, 17, 10], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[4]:=	GaussCode[Link[9, Alternating, 50]]
Out[4]=	GaussCode[{1, -8, 2, -9}, {8, -1, 4, -6, 7, -5}, {5, -2, 9, -4, 3, -7, 6, -3}]
In[5]:=	BR[Link[9, Alternating, 50]]
Out[5]=	BR[Link[9, Alternating, 50]]
In[6]:=	alex = Alexander[Link[9, Alternating, 50]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 50]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 50]], KnotSignature[Link[9, Alternating, 50]]}
Out[9]=	{Infinity, 2}
In[10]:=	J=Jones[Link[9, Alternating, 50]][q]
Out[10]=	-3 2 5 2 3 4 5 6 -5 + q - -- + - + 8 q - 7 q + 7 q - 5 q + 3 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 50]][q]
Out[12]=	-10 -8 -6 4 3 2 4 6 10 14 5 + q + q + q + -- + -- + 5 q + 2 q + 4 q + 2 q - q + 4 2 q q 16 18 q - q
In[13]:=	Kauffman[Link[9, Alternating, 50]][a, z]
Out[13]=	2 2 8 2 2 1 a 2 2 a z 3 z 5 z -9 - -- - -- - 4 a + -- + ----- + -- - --- - --- + -- + --- + --- + 4 2 2 2 2 2 a z z 5 3 a a a z a z z a a 2 2 2 3 3 3 2 z 6 z 23 z 2 2 z 4 z 5 z 3 a z + 22 z - -- + ---- + ----- + 6 a z + -- - ---- - ---- + 6 4 2 7 5 3 a a a a a a 3 4 4 4 5 5 3 z 3 4 3 z 9 z 27 z 2 4 5 z 3 z ---- + 3 a z - 19 z + ---- - ---- - ----- - 4 a z + ---- - ---- - a 6 4 2 5 3 a a a a a 5 6 6 7 7 14 z 5 6 6 z 8 z 2 6 4 z 6 z 7 ----- - 6 a z + 3 z + ---- + ---- + a z + ---- + ---- + 2 a z + a 4 2 3 a a a a 8 8 z z + -- 2 a
In[14]:=	{Vassiliev[2][Link[9, Alternating, 50]], Vassiliev[3][Link[9, Alternating, 50]]}
Out[14]=	{0, 1}
In[15]:=	Kh[Link[9, Alternating, 50]][q, t]
Out[15]=	3 1 1 1 4 2 2 3 q 6 q + 4 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 3 q t + 4 q t + 4 q t + 4 q t + 2 q t + 3 q t + q t + 11 4 13 5 2 q t + q t