L9a7

L9a6

L9a8

	Visit L9a7's page at Knotilus! Visit L9a7's page at the original Knot Atlas!
	L9a7 is [math]\displaystyle{ 9^2_{17} }[/math] in the Rolfsen table of links.

L9a7 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ -\frac{3 u v^2-4 u v+2 u+2 v^3-4 v^2+3 v}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{5}{q^{9/2}}-\frac{4}{q^{7/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{2}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{5}{q^{15/2}}+\frac{5}{q^{13/2}}-\frac{7}{q^{11/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{11} z^{-1} +3 a^9 z+2 a^9 z^{-1} -2 a^7 z^3-2 a^7 z-2 a^5 z^3-2 a^5 z-a^5 z^{-1} -a^3 z^3-a^3 z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^{12}+4 z^4 a^{12}-5 z^2 a^{12}+2 a^{12}-2 z^7 a^{11}+7 z^5 a^{11}-6 z^3 a^{11}+2 z a^{11}-a^{11} z^{-1} -z^8 a^{10}-z^6 a^{10}+13 z^4 a^{10}-15 z^2 a^{10}+5 a^{10}-5 z^7 a^9+15 z^5 a^9-12 z^3 a^9+5 z a^9-2 a^9 z^{-1} -z^8 a^8-3 z^6 a^8+13 z^4 a^8-10 z^2 a^8+3 a^8-3 z^7 a^7+5 z^5 a^7-2 z^3 a^7-z a^7-3 z^6 a^6+2 z^4 a^6+z^2 a^6-a^6-3 z^5 a^5+3 z^3 a^5-3 z a^5+a^5 z^{-1} -2 z^4 a^4+z^2 a^4-z^3 a^3+z a^3 }[/math] (db)

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ \frac{211}{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:L9a7/V 2,1 Data:L9a7/V 3,1 Data:L9a7/V 4,1 Data:L9a7/V 4,2 Data:L9a7/V 4,3 Data:L9a7/V 5,1 Data:L9a7/V 5,2 Data:L9a7/V 5,3 Data:L9a7/V 5,4 Data:L9a7/V 6,1 Data:L9a7/V 6,2 Data:L9a7/V 6,3 Data:L9a7/V 6,4 Data:L9a7/V 6,5 Data:L9a7/V 6,6 Data:L9a7/V 6,7 Data:L9a7/V 6,8 Data:L9a7/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 L9a7. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                   1 1 -4                 2 1 -1 -6               2     2 -8             3 2     -1 -10           4 2       2 -12         2 4         2 -14       3 3           0 -16     1 2             1 -18   1 3               -2 -20   1                 1 -22 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=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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, 7]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 7]]
Out[3]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 13, 17, 14], X[14, 7, 15, 8], X[8, 15, 9, 16], X[18, 11, 5, 12], X[12, 17, 13, 18], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[4]:=	GaussCode[Link[9, Alternating, 7]]
Out[4]=	GaussCode[{1, -8, 2, -9}, {8, -1, 4, -5, 9, -2, 6, -7, 3, -4, 5, -3, 7, -6}]
In[5]:=	BR[Link[9, Alternating, 7]]
Out[5]=	BR[Link[9, Alternating, 7]]
In[6]:=	alex = Alexander[Link[9, Alternating, 7]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 7]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 7]], KnotSignature[Link[9, Alternating, 7]]}
Out[9]=	{Infinity, -3}
In[10]:=	J=Jones[Link[9, Alternating, 7]][q]
Out[10]=	-(21/2) 2 4 5 5 7 5 4 2 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q q -(3/2) q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 7]][q]
Out[12]=	-34 2 -28 -26 2 -22 3 3 2 2 -12 -q - --- - q - q + --- + q + --- + --- + --- + --- - q + 32 24 20 18 16 14 q q q q q q -10 -8 -6 -4 q + q - q + q
In[13]:=	Kauffman[Link[9, Alternating, 7]][a, z]
Out[13]=	5 9 11 6 8 10 12 a 2 a a 3 5 7 -a + 3 a + 5 a + 2 a + -- - ---- - --- + a z - 3 a z - a z + z z z 9 11 4 2 6 2 8 2 10 2 12 2 5 a z + 2 a z + a z + a z - 10 a z - 15 a z - 5 a z - 3 3 5 3 7 3 9 3 11 3 4 4 6 4 a z + 3 a z - 2 a z - 12 a z - 6 a z - 2 a z + 2 a z + 8 4 10 4 12 4 5 5 7 5 9 5 13 a z + 13 a z + 4 a z - 3 a z + 5 a z + 15 a z + 11 5 6 6 8 6 10 6 12 6 7 7 9 7 7 a z - 3 a z - 3 a z - a z - a z - 3 a z - 5 a z - 11 7 8 8 10 8 2 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 7]], Vassiliev[3][Link[9, Alternating, 7]]}
Out[14]=	211 {0, ---} 24
In[15]:=	Kh[Link[9, Alternating, 7]][q, t]
Out[15]=	-4 -2 1 1 1 3 1 2 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 22 9 20 8 18 8 18 7 16 7 16 6 q t q t q t q t q t q t 3 3 2 4 4 2 3 2 ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 14 6 14 5 12 5 12 4 10 4 10 3 8 3 8 2 q t q t q t q t q t q t q t q t 2 2 ----- + ---- 6 2 4 q t q t