L9a48

L9a47

L9a49

	Visit L9a48's page at Knotilus! Visit L9a48's page at the original Knot Atlas!
	L9a48 is [math]\displaystyle{ 9^3_{5} }[/math] in the Rolfsen table of links.

L9a48 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ \frac{u v^2 w^2-u v^2 w-u v w^2+2 u v w-u v-u w+2 u-2 v^2 w^2+v^2 w+v w^2-2 v w+v+w-1}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^6+3 q^5-4 q^4+6 q^3-6 q^2+6 q-4+4 q^{-1} - q^{-2} + q^{-3} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^{-4} -2 z^2 a^{-4} +z^6 a^{-2} +4 z^4 a^{-2} +a^2 z^2+5 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +3 a^2+3 a^{-2} -2 z^4-7 z^2-2 z^{-2} -6 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^8 a^{-2} +z^8+a z^7+4 z^7 a^{-1} +3 z^7 a^{-3} +a^2 z^6+2 z^6 a^{-2} +4 z^6 a^{-4} -z^6-2 a z^5-10 z^5 a^{-1} -4 z^5 a^{-3} +4 z^5 a^{-5} -5 a^2 z^4-8 z^4 a^{-2} -4 z^4 a^{-4} +3 z^4 a^{-6} -6 z^4-3 a z^3+2 z^3 a^{-1} +z^3 a^{-3} -3 z^3 a^{-5} +z^3 a^{-7} +8 a^2 z^2+5 z^2 a^{-2} -2 z^2 a^{-6} +11 z^2+6 a z+6 z a^{-1} -5 a^2-3 a^{-2} + a^{-4} -8-2 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, 0)

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:L9a48/V 2,1 Data:L9a48/V 3,1 Data:L9a48/V 4,1 Data:L9a48/V 4,2 Data:L9a48/V 4,3 Data:L9a48/V 5,1 Data:L9a48/V 5,2 Data:L9a48/V 5,3 Data:L9a48/V 5,4 Data:L9a48/V 6,1 Data:L9a48/V 6,2 Data:L9a48/V 6,3 Data:L9a48/V 6,4 Data:L9a48/V 6,5 Data:L9a48/V 6,6 Data:L9a48/V 6,7 Data:L9a48/V 6,8 Data:L9a48/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 L9a48. 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               2 1   -1 7             4 2     2 5           4 4       0 3         2 2         0 1       3 5           2 -1     1 1             0 -3     3               3 -5 1 1                 0 -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{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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, 48]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 48]]
Out[3]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[18, 14, 11, 13], X[8, 16, 9, 15], X[14, 8, 15, 7], X[16, 10, 17, 9], X[10, 18, 5, 17], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[4]:=	GaussCode[Link[9, Alternating, 48]]
Out[4]=	GaussCode[{1, -8, 2, -9}, {8, -1, 5, -4, 6, -7}, {9, -2, 3, -5, 4, -6, 7, -3}]
In[5]:=	BR[Link[9, Alternating, 48]]
Out[5]=	BR[Link[9, Alternating, 48]]
In[6]:=	alex = Alexander[Link[9, Alternating, 48]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 48]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 48]], KnotSignature[Link[9, Alternating, 48]]}
Out[9]=	{Infinity, 2}
In[10]:=	J=Jones[Link[9, Alternating, 48]][q]
Out[10]=	-3 -2 4 2 3 4 5 6 -4 + q - q + - + 6 q - 6 q + 6 q - 4 q + 3 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 48]][q]
Out[12]=	-10 2 2 4 3 2 4 6 10 12 16 4 + q + -- + -- + -- + -- + 4 q + q + 3 q + 2 q + q + q - 8 6 4 2 q q q q 18 q
In[13]:=	Kauffman[Link[9, Alternating, 48]][a, z]
Out[13]=	2 -4 3 2 2 1 a 2 2 a 6 z -8 + a - -- - 5 a + -- + ----- + -- - --- - --- + --- + 6 a z + 2 2 2 2 2 a z z a a z a z z 2 2 3 3 3 3 2 2 z 5 z 2 2 z 3 z z 2 z 3 11 z - ---- + ---- + 8 a z + -- - ---- + -- + ---- - 3 a z - 6 2 7 5 3 a a a a a a 4 4 4 5 5 5 4 3 z 4 z 8 z 2 4 4 z 4 z 10 z 5 6 z + ---- - ---- - ---- - 5 a z + ---- - ---- - ----- - 2 a z - 6 4 2 5 3 a a a a a a 6 6 7 7 8 6 4 z 2 z 2 6 3 z 4 z 7 8 z z + ---- + ---- + a z + ---- + ---- + a z + z + -- 4 2 3 a 2 a a a a
In[14]:=	{Vassiliev[2][Link[9, Alternating, 48]], Vassiliev[3][Link[9, Alternating, 48]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Link[9, Alternating, 48]][q, t]
Out[15]=	3 1 1 1 3 1 1 3 q 5 q + 2 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 7 4 5 4 5 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 2 q t + 4 q t + 4 q t + 4 q t + 2 q t + 2 q t + q t + 11 4 13 5 2 q t + q t