L9a27

L9a26

L9a28

	Visit L9a27's page at Knotilus! Visit L9a27's page at the original Knot Atlas!
	L9a27 is [math]\displaystyle{ 9^2_{12} }[/math] in the Rolfsen table of links.

L9a27 Further Notes and Views

Knot presentations

Planar diagram presentation	X8192 X10,4,11,3 X18,10,7,9 X2738 X16,13,17,14 X6,12,1,11 X4,16,5,15 X14,6,15,5 X12,17,13,18
Gauss code	{1, -4, 2, -7, 8, -6}, {4, -1, 3, -2, 6, -9, 5, -8, 7, -5, 9, -3}

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ -\frac{31}{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:L9a27/V 2,1 Data:L9a27/V 3,1 Data:L9a27/V 4,1 Data:L9a27/V 4,2 Data:L9a27/V 4,3 Data:L9a27/V 5,1 Data:L9a27/V 5,2 Data:L9a27/V 5,3 Data:L9a27/V 5,4 Data:L9a27/V 6,1 Data:L9a27/V 6,2 Data:L9a27/V 6,3 Data:L9a27/V 6,4 Data:L9a27/V 6,5 Data:L9a27/V 6,6 Data:L9a27/V 6,7 Data:L9a27/V 6,8 Data:L9a27/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 L9a27. 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               3 1   -2 6             5 2     3 4           4 4       0 2         5 4         1 0       3 5           2 -2     2 4             -2 -4   1 3               2 -6   2                 -2 -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{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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, 27]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 27]]
Out[3]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[18, 10, 7, 9], X[2, 7, 3, 8], X[16, 13, 17, 14], X[6, 12, 1, 11], X[4, 16, 5, 15], X[14, 6, 15, 5], X[12, 17, 13, 18]]
In[4]:=	GaussCode[Link[9, Alternating, 27]]
Out[4]=	GaussCode[{1, -4, 2, -7, 8, -6}, {4, -1, 3, -2, 6, -9, 5, -8, 7, -5, 9, -3}]
In[5]:=	BR[Link[9, Alternating, 27]]
Out[5]=	BR[Link[9, Alternating, 27]]
In[6]:=	alex = Alexander[Link[9, Alternating, 27]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 27]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 27]], KnotSignature[Link[9, Alternating, 27]]}
Out[9]=	{Infinity, 1}
In[10]:=	J=Jones[Link[9, Alternating, 27]][q]
Out[10]=	-(7/2) 3 5 7 3/2 5/2 -q + ---- - ---- + ------- - 9 Sqrt[q] + 8 q - 8 q + 5/2 3/2 Sqrt[q] q q 7/2 9/2 11/2 5 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, 27]][q]
Out[12]=	-12 2 -6 -4 4 8 10 12 14 18 3 + q - -- + q - q + 3 q + 2 q + 2 q - q + 2 q - q 8 q
In[13]:=	Kauffman[Link[9, Alternating, 27]][a, z]
Out[13]=	2 2 -2 1 1 2 z 4 z 2 z 3 2 z z -a + ---- + --- - --- - --- - --- - a z - a z - 7 z + -- - -- - 3 a z 5 3 a 6 4 a z a a a a 2 3 3 3 4 5 z 2 2 4 z 6 z z 3 3 3 4 z ---- - 4 a z + ---- + ---- + -- + a z + 2 a z + 15 z - -- + 2 5 3 a 6 a a a a 4 4 5 5 5 4 z 13 z 2 4 3 z z 7 z 5 3 5 6 ---- + ----- + 7 a z - ---- - -- + ---- + 4 a z - a z - 6 z - 4 2 5 3 a a a a a 6 6 7 7 8 4 z 7 z 2 6 3 z 6 z 7 8 z ---- - ---- - 3 a z - ---- - ---- - 3 a z - z - -- 4 2 3 a 2 a a a a
In[14]:=	{Vassiliev[2][Link[9, Alternating, 27]], Vassiliev[3][Link[9, Alternating, 27]]}
Out[14]=	31 {0, -(--)} 48
In[15]:=	Kh[Link[9, Alternating, 27]][q, t]
Out[15]=	2 1 2 1 3 2 3 4 2 5 + 5 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 4 q t + 8 4 6 3 4 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 4 q t + 4 q t + 5 q t + 2 q t + 3 q t + q t + 2 q t + 12 5 q t