L9a16

L9a15

L9a17

	Visit L9a16's page at Knotilus! Visit L9a16's page at the original Knot Atlas!
	L9a16 is [math]\displaystyle{ 9^2_{28} }[/math] in the Rolfsen table of links.

L9a16 Further Notes and Views

Knot presentations

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

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ \frac{55}{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:L9a16/V 2,1 Data:L9a16/V 3,1 Data:L9a16/V 4,1 Data:L9a16/V 4,2 Data:L9a16/V 4,3 Data:L9a16/V 5,1 Data:L9a16/V 5,2 Data:L9a16/V 5,3 Data:L9a16/V 5,4 Data:L9a16/V 6,1 Data:L9a16/V 6,2 Data:L9a16/V 6,3 Data:L9a16/V 6,4 Data:L9a16/V 6,5 Data:L9a16/V 6,6 Data:L9a16/V 6,7 Data:L9a16/V 6,8 Data:L9a16/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 L9a16. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                   1 1 0                 2   -2 -2               3 1   2 -4             4 3     -1 -6           4 2       2 -8         3 4         1 -10       4 4           0 -12     1 4             3 -14   1 3               -2 -16   1                 1 -18 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/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}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/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, 16]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 16]]
Out[3]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[14, 10, 15, 9], X[10, 14, 11, 13], X[18, 15, 5, 16], X[16, 7, 17, 8], X[8, 17, 9, 18], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[4]:=	GaussCode[Link[9, Alternating, 16]]
Out[4]=	GaussCode[{1, -8, 2, -9}, {8, -1, 6, -7, 3, -4, 9, -2, 4, -3, 5, -6, 7, -5}]
In[5]:=	BR[Link[9, Alternating, 16]]
Out[5]=	BR[Link[9, Alternating, 16]]
In[6]:=	alex = Alexander[Link[9, Alternating, 16]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 16]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 16]], KnotSignature[Link[9, Alternating, 16]]}
Out[9]=	{Infinity, -3}
In[10]:=	J=Jones[Link[9, Alternating, 16]][q]
Out[10]=	-(17/2) 2 4 7 7 8 6 5 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q 3 ------- - Sqrt[q] Sqrt[q]
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 16]][q]
Out[12]=	-28 2 4 2 3 2 2 2 -4 2 -1 - q - --- + --- + --- + --- + --- + -- - -- + q + q 26 18 16 14 12 8 6 q q q q q q q
In[13]:=	Kauffman[Link[9, Alternating, 16]][a, z]
Out[13]=	5 7 9 6 8 10 2 a 3 a a 5 7 -3 a - 3 a - a + ---- + ---- + -- - a z - 4 a z - 7 a z - z z z 9 2 2 4 2 6 2 8 2 10 2 3 2 a z - 3 a z - 5 a z + a z + 5 a z + 2 a z + 2 a z + 7 3 9 3 2 4 4 4 6 4 8 4 10 4 5 a z + 3 a z + 7 a z + 11 a z + 4 a z - a z - a z - 5 3 5 5 5 7 5 9 5 2 6 4 6 a z + 5 a z + 7 a z - a z - 2 a z - 3 a z - 4 a z - 6 6 8 6 3 7 5 7 7 7 4 8 6 8 3 a z - 2 a z - 3 a z - 5 a z - 2 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 16]], Vassiliev[3][Link[9, Alternating, 16]]}
Out[14]=	55 {0, --} 24
In[15]:=	Kh[Link[9, Alternating, 16]][q, t]
Out[15]=	3 3 1 1 1 3 1 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 q q q t q t q t q t q t q t 4 4 3 4 4 2 4 t ------ + ------ + ----- + ----- + ----- + ---- + ---- + 2 t + -- + 10 4 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q t q 2 2 q t