L9a28

L9a27

L9a29

	Visit L9a28's page at Knotilus! Visit L9a28's page at the original Knot Atlas!
	L9a28 is [math]\displaystyle{ 9^2_{20} }[/math] in the Rolfsen table of links.

L9a28 Further Notes and Views

Knot presentations

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X18,16,7,15 X16,12,17,11 X12,18,13,17 X6718 X4,13,5,14
Gauss code	{1, -2, 3, -9, 4, -8}, {8, -1, 2, -3, 6, -7, 9, -4, 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{t(1)^2 t(2)^4-2 t(1)^2 t(2)^3+2 t(1) t(2)^3+2 t(1)^2 t(2)^2-3 t(1) t(2)^2+2 t(2)^2+2 t(1) t(2)-2 t(2)+1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{5}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{5}{q^{5/2}}+q^{3/2}-\frac{5}{q^{3/2}}-\frac{1}{q^{15/2}}+\frac{2}{q^{13/2}}-\frac{4}{q^{11/2}}-2 \sqrt{q}+\frac{3}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^5+4 a^5 z^3+5 a^5 z+2 a^5 z^{-1} -a^3 z^7-6 a^3 z^5-13 a^3 z^3-12 a^3 z-3 a^3 z^{-1} +a z^5+4 a z^3+4 a z+a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^3-a^9 z+2 a^8 z^4-a^8 z^2+3 a^7 z^5-3 a^7 z^3+2 a^7 z+3 a^6 z^6-3 a^6 z^4+a^6 z^2+3 a^5 z^7-7 a^5 z^5+9 a^5 z^3-7 a^5 z+2 a^5 z^{-1} +a^4 z^8+2 a^4 z^6-11 a^4 z^4+10 a^4 z^2-3 a^4+5 a^3 z^7-18 a^3 z^5+22 a^3 z^3-14 a^3 z+3 a^3 z^{-1} +a^2 z^8-10 a^2 z^4+12 a^2 z^2-3 a^2+2 a z^7-8 a z^5+9 a z^3-4 a z+a z^{-1} +z^6-4 z^4+4 z^2-1 }[/math] (db)

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ -\frac{121}{16} }[/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:L9a28/V 2,1 Data:L9a28/V 3,1 Data:L9a28/V 4,1 Data:L9a28/V 4,2 Data:L9a28/V 4,3 Data:L9a28/V 5,1 Data:L9a28/V 5,2 Data:L9a28/V 5,3 Data:L9a28/V 5,4 Data:L9a28/V 6,1 Data:L9a28/V 6,2 Data:L9a28/V 6,3 Data:L9a28/V 6,4 Data:L9a28/V 6,5 Data:L9a28/V 6,6 Data:L9a28/V 6,7 Data:L9a28/V 6,8 Data:L9a28/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 L9a28. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                   1 -1 2                 1   1 0               2 1   -1 -2             3 1     2 -4           3 3       0 -6         3 2         1 -8       2 3           1 -10     2 3             -1 -12     2               2 -14 1 2                 -1 -16 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/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, 28]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 28]]
Out[3]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[14, 5, 15, 6], X[18, 16, 7, 15], X[16, 12, 17, 11], X[12, 18, 13, 17], X[6, 7, 1, 8], X[4, 13, 5, 14]]
In[4]:=	GaussCode[Link[9, Alternating, 28]]
Out[4]=	GaussCode[{1, -2, 3, -9, 4, -8}, {8, -1, 2, -3, 6, -7, 9, -4, 5, -6, 7, -5}]
In[5]:=	BR[Link[9, Alternating, 28]]
Out[5]=	BR[Link[9, Alternating, 28]]
In[6]:=	alex = Alexander[Link[9, Alternating, 28]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 28]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 28]], KnotSignature[Link[9, Alternating, 28]]}
Out[9]=	{Infinity, -3}
In[10]:=	J=Jones[Link[9, Alternating, 28]][q]
Out[10]=	-(15/2) 2 4 5 6 5 5 3 -q + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 2 Sqrt[q] + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 28]][q]
Out[12]=	-22 2 -16 -14 2 3 -4 4 -1 + q + --- + q + q + --- + -- + q - q 18 12 8 q q q
In[13]:=	Kauffman[Link[9, Alternating, 28]][a, z]
Out[13]=	3 5 2 4 a 3 a 2 a 3 5 7 1 + 3 a + 3 a - - - ---- - ---- + 4 a z + 14 a z + 7 a z - 2 a z + z z z 9 2 2 2 4 2 6 2 8 2 3 a z - 4 z - 12 a z - 10 a z - a z + a z - 9 a z - 3 3 5 3 7 3 9 3 4 2 4 4 4 22 a z - 9 a z + 3 a z - a z + 4 z + 10 a z + 11 a z + 6 4 8 4 5 3 5 5 5 7 5 6 3 a z - 2 a z + 8 a z + 18 a z + 7 a z - 3 a z - z - 4 6 6 6 7 3 7 5 7 2 8 4 8 2 a z - 3 a z - 2 a z - 5 a z - 3 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 28]], Vassiliev[3][Link[9, Alternating, 28]]}
Out[14]=	121 {0, -(---)} 16
In[15]:=	Kh[Link[9, Alternating, 28]][q, t]
Out[15]=	3 3 1 1 2 2 2 3 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 4 2 16 6 14 6 14 5 12 4 10 4 10 3 8 3 q q q t q t q t q t q t q t q t 3 3 2 3 t 2 2 2 4 3 ----- + ----- + ---- + ---- + 2 t + -- + t + q t + q t 8 2 6 2 6 4 2 q t q t q t q t q