L9a24

L9a23

L9a25

	Visit L9a24's page at Knotilus! Visit L9a24's page at the original Knot Atlas!
	L9a24 is [math]\displaystyle{ 9^2_{21} }[/math] in the Rolfsen table of links.

L9a24 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ \frac{u^2 v^3-u^2 v^2+u^2 v-u^2+u v^4-3 u v^3+3 u v^2-3 u v+u-v^4+v^3-v^2+v}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\sqrt{q}+\frac{2}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{6}{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{ -z^3 a^7-2 z a^7-a^7 z^{-1} +z^5 a^5+3 z^3 a^5+4 z a^5+3 a^5 z^{-1} +z^5 a^3+2 z^3 a^3-z a^3-2 a^3 z^{-1} -z^3 a-2 z a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^4-2 a^{10} z^2+2 a^9 z^5-3 a^9 z^3+3 a^8 z^6-6 a^8 z^4+5 a^8 z^2-a^8+3 a^7 z^7-8 a^7 z^5+13 a^7 z^3-7 a^7 z+a^7 z^{-1} +a^6 z^8+2 a^6 z^6-8 a^6 z^4+11 a^6 z^2-3 a^6+5 a^5 z^7-15 a^5 z^5+22 a^5 z^3-14 a^5 z+3 a^5 z^{-1} +a^4 z^8+a^4 z^6-6 a^4 z^4+6 a^4 z^2-3 a^4+2 a^3 z^7-4 a^3 z^5+3 a^3 z^3-5 a^3 z+2 a^3 z^{-1} +2 a^2 z^6-5 a^2 z^4+2 a^2 z^2+a z^5-3 a z^3+2 a z }[/math] (db)

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ \frac{247}{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:L9a24/V 2,1 Data:L9a24/V 3,1 Data:L9a24/V 4,1 Data:L9a24/V 4,2 Data:L9a24/V 4,3 Data:L9a24/V 5,1 Data:L9a24/V 5,2 Data:L9a24/V 5,3 Data:L9a24/V 5,4 Data:L9a24/V 6,1 Data:L9a24/V 6,2 Data:L9a24/V 6,3 Data:L9a24/V 6,4 Data:L9a24/V 6,5 Data:L9a24/V 6,6 Data:L9a24/V 6,7 Data:L9a24/V 6,8 Data:L9a24/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 L9a24. 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                 1   -1 -2               3 1   2 -4             3 2     -1 -6           4 2       2 -8         3 4         1 -10       3 3           0 -12     1 3             2 -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}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{2}\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 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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, 24]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 24]]
Out[3]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[14, 6, 15, 5], X[16, 11, 17, 12], X[18, 13, 7, 14], X[12, 17, 13, 18], X[4, 16, 5, 15], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[4]:=	GaussCode[Link[9, Alternating, 24]]
Out[4]=	GaussCode[{1, -8, 2, -7, 3, -9}, {8, -1, 9, -2, 4, -6, 5, -3, 7, -4, 6, -5}]
In[5]:=	BR[Link[9, Alternating, 24]]
Out[5]=	BR[Link[9, Alternating, 24]]
In[6]:=	alex = Alexander[Link[9, Alternating, 24]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 24]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 24]], KnotSignature[Link[9, Alternating, 24]]}
Out[9]=	{Infinity, -3}
In[10]:=	J=Jones[Link[9, Alternating, 24]][q]
Out[10]=	-(17/2) 2 4 6 6 7 5 4 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q 2 ------- - 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, 24]][q]
Out[12]=	-26 -22 2 -18 3 3 2 3 -6 -4 2 -q - q - --- + q + --- + --- + --- + -- - q + q + q 20 14 12 10 8 q q q q q
In[13]:=	Kauffman[Link[9, Alternating, 24]][a, z]
Out[13]=	3 5 7 4 6 8 2 a 3 a a 3 5 3 a + 3 a + a - ---- - ---- - -- - 2 a z + 5 a z + 14 a z + z z z 7 2 2 4 2 6 2 8 2 10 2 3 7 a z - 2 a z - 6 a z - 11 a z - 5 a z + 2 a z + 3 a z - 3 3 5 3 7 3 9 3 2 4 4 4 3 a z - 22 a z - 13 a z + 3 a z + 5 a z + 6 a z + 6 4 8 4 10 4 5 3 5 5 5 7 5 8 a z + 6 a z - a z - a z + 4 a z + 15 a z + 8 a z - 9 5 2 6 4 6 6 6 8 6 3 7 5 7 2 a z - 2 a z - a z - 2 a z - 3 a z - 2 a z - 5 a z - 7 7 4 8 6 8 3 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 24]], Vassiliev[3][Link[9, Alternating, 24]]}
Out[14]=	247 {0, ---} 48
In[15]:=	Kh[Link[9, Alternating, 24]][q, t]
Out[15]=	2 3 1 1 1 3 1 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + 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 3 3 3 4 4 2 3 t 2 2 ------ + ------ + ----- + ----- + ----- + ---- + ---- + t + -- + q 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