L9a14

L9a13

L9a15

	Visit L9a14's page at Knotilus! Visit L9a14's page at the original Knot Atlas!
	L9a14 is [math]\displaystyle{ 9^2_{13} }[/math] in the Rolfsen table of links.

L9a14 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ -\frac{(u-1) (v-1) \left(v^2-v+1\right) \left(v^2+v+1\right)}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -4 q^{9/2}+2 q^{7/2}-3 q^{5/2}+q^{3/2}+q^{19/2}-2 q^{17/2}+3 q^{15/2}-3 q^{13/2}+4 q^{11/2}-\sqrt{q} }[/math] (db)
Signature	5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^{-7} +4 z^3 a^{-7} +4 z a^{-7} +2 a^{-7} z^{-1} -z^7 a^{-5} -6 z^5 a^{-5} -12 z^3 a^{-5} -11 z a^{-5} -5 a^{-5} z^{-1} +z^5 a^{-3} +5 z^3 a^{-3} +7 z a^{-3} +3 a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^8 a^{-4} -z^8 a^{-6} -z^7 a^{-3} -4 z^7 a^{-5} -3 z^7 a^{-7} +4 z^6 a^{-4} +z^6 a^{-6} -3 z^6 a^{-8} +6 z^5 a^{-3} +20 z^5 a^{-5} +11 z^5 a^{-7} -3 z^5 a^{-9} -z^4 a^{-4} +8 z^4 a^{-6} +6 z^4 a^{-8} -3 z^4 a^{-10} -12 z^3 a^{-3} -30 z^3 a^{-5} -12 z^3 a^{-7} +4 z^3 a^{-9} -2 z^3 a^{-11} -8 z^2 a^{-4} -12 z^2 a^{-6} +3 z^2 a^{-10} -z^2 a^{-12} +10 z a^{-3} +17 z a^{-5} +7 z a^{-7} +5 a^{-4} +5 a^{-6} - a^{-10} -3 a^{-3} z^{-1} -5 a^{-5} z^{-1} -2 a^{-7} z^{-1} }[/math] (db)

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ -\frac{23}{2} }[/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:L9a14/V 2,1 Data:L9a14/V 3,1 Data:L9a14/V 4,1 Data:L9a14/V 4,2 Data:L9a14/V 4,3 Data:L9a14/V 5,1 Data:L9a14/V 5,2 Data:L9a14/V 5,3 Data:L9a14/V 5,4 Data:L9a14/V 6,1 Data:L9a14/V 6,2 Data:L9a14/V 6,3 Data:L9a14/V 6,4 Data:L9a14/V 6,5 Data:L9a14/V 6,6 Data:L9a14/V 6,7 Data:L9a14/V 6,8 Data:L9a14/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]5 is the signature of L9a14. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 χ 20                   1 -1 18                 1   1 16               2 1   -1 14             1 1     0 12           3 2       -1 10         1 1         0 8       1 3           2 6     2 1             1 4   1 3               2 2                     0 0 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=6 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=7 }[/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, 14]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 14]]
Out[3]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 8, 15, 7], X[16, 10, 17, 9], X[18, 12, 5, 11], X[8, 16, 9, 15], X[10, 18, 11, 17], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[4]:=	GaussCode[Link[9, Alternating, 14]]
Out[4]=	GaussCode[{1, -8, 2, -9}, {8, -1, 3, -6, 4, -7, 5, -2, 9, -3, 6, -4, 7, -5}]
In[5]:=	BR[Link[9, Alternating, 14]]
Out[5]=	BR[Link[9, Alternating, 14]]
In[6]:=	alex = Alexander[Link[9, Alternating, 14]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 14]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 14]], KnotSignature[Link[9, Alternating, 14]]}
Out[9]=	{Infinity, 5}
In[10]:=	J=Jones[Link[9, Alternating, 14]][q]
Out[10]=	3/2 5/2 7/2 9/2 11/2 13/2 -Sqrt[q] + q - 3 q + 2 q - 4 q + 4 q - 3 q + 15/2 17/2 19/2 3 q - 2 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 14]][q]
Out[12]=	2 4 6 8 10 12 14 16 18 20 q + q + 2 q + 3 q + 3 q + 4 q + q + q - 2 q - 2 q - 22 24 28 q - q - q
In[13]:=	Kauffman[Link[9, Alternating, 14]][a, z]
Out[13]=	2 2 -10 5 5 2 5 3 7 z 17 z 10 z z 3 z -a + -- + -- - ---- - ---- - ---- + --- + ---- + ---- - --- + ---- - 6 4 7 5 3 7 5 3 12 10 a a a z a z a z a a a a a 2 2 3 3 3 3 3 4 4 12 z 8 z 2 z 4 z 12 z 30 z 12 z 3 z 6 z ----- - ---- - ---- + ---- - ----- - ----- - ----- - ---- + ---- + 6 4 11 9 7 5 3 10 8 a a a a a a a a a 4 4 5 5 5 5 6 6 6 7 8 z z 3 z 11 z 20 z 6 z 3 z z 4 z 3 z ---- - -- - ---- + ----- + ----- + ---- - ---- + -- + ---- - ---- - 6 4 9 7 5 3 8 6 4 7 a a a a a a a a a a 7 7 8 8 4 z z z z ---- - -- - -- - -- 5 3 6 4 a a a a
In[14]:=	{Vassiliev[2][Link[9, Alternating, 14]], Vassiliev[3][Link[9, Alternating, 14]]}
Out[14]=	23 {0, -(--)} 2
In[15]:=	Kh[Link[9, Alternating, 14]][q, t]
Out[15]=	4 4 6 -2 q 6 8 8 2 10 2 10 3 3 q + 2 q + t + -- + q t + q t + 3 q t + q t + q t + t 12 3 12 4 14 4 14 5 16 5 16 6 18 6 3 q t + 2 q t + q t + q t + 2 q t + q t + q t + 20 7 q t