L6a5

L6a4

L6n1

	Visit L6a5's page at Knotilus! Visit L6a5's page at the original Knot Atlas!
	L6a5 is [math]\displaystyle{ 6^3_1 }[/math] in the Rolfsen table of links. It is a closed three-link chain.

Stained glass window of Trinity symbol, Brazil

French coat of arms.

Russian coat of arms.

Russian passport page-number decoration.

Knot presentations

Planar diagram presentation	X6172 X10,3,11,4 X12,7,9,8 X8,11,5,12 X2536 X4,9,1,10
Gauss code	{1, -5, 2, -6}, {5, -1, 3, -4}, {6, -2, 4, -3}

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ \frac{27}{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:L6a5/V 2,1 Data:L6a5/V 3,1 Data:L6a5/V 4,1 Data:L6a5/V 4,2 Data:L6a5/V 4,3 Data:L6a5/V 5,1 Data:L6a5/V 5,2 Data:L6a5/V 5,3 Data:L6a5/V 5,4 Data:L6a5/V 6,1 Data:L6a5/V 6,2 Data:L6a5/V 6,3 Data:L6a5/V 6,4 Data:L6a5/V 6,5 Data:L6a5/V 6,6 Data:L6a5/V 6,7 Data:L6a5/V 6,8 Data:L6a5/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]-2 is the signature of L6a5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 χ -1             1 1 -3           2 1 -1 -5         1     1 -7         2     2 -9     3 1       2 -11   1 3         2 -13               0 -15 1             1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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[6, Alternating, 5]]
Out[2]=	6
In[3]:=	PD[Link[6, Alternating, 5]]
Out[3]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[12, 7, 9, 8], X[8, 11, 5, 12], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[4]:=	GaussCode[Link[6, Alternating, 5]]
Out[4]=	GaussCode[{1, -5, 2, -6}, {5, -1, 3, -4}, {6, -2, 4, -3}]
In[5]:=	BR[Link[6, Alternating, 5]]
Out[5]=	BR[Link[6, Alternating, 5]]
In[6]:=	alex = Alexander[Link[6, Alternating, 5]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[6, Alternating, 5]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[6, Alternating, 5]], KnotSignature[Link[6, Alternating, 5]]}
Out[9]=	{Infinity, -2}
In[10]:=	J=Jones[Link[6, Alternating, 5]][q]
Out[10]=	-7 -6 3 -4 3 2 1 q - q + -- - q + -- - -- + - 5 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[6, Alternating, 5]][q]
Out[12]=	-24 3 3 4 5 4 4 2 -8 -4 -2 q + --- + --- + --- + --- + --- + --- + --- + q - q + q 22 20 18 16 14 12 10 q q q q q q q
In[13]:=	Kauffman[Link[6, Alternating, 5]][a, z]
Out[13]=	4 6 8 5 7 4 6 8 a 2 a a 2 a 2 a 5 7 3 a + 5 a + 3 a - -- - ---- - -- + ---- + ---- - 3 a z - 3 a z + 2 2 2 z z z z z 2 2 4 2 6 2 8 2 3 3 5 3 7 3 a z - 5 a z - 9 a z - 3 a z + 2 a z + a z - a z + 4 4 6 4 8 4 5 5 7 5 3 a z + 4 a z + a z + a z + a z
In[14]:=	{Vassiliev[2][Link[6, Alternating, 5]], Vassiliev[3][Link[6, Alternating, 5]]}
Out[14]=	27 {0, --} 2
In[15]:=	Kh[Link[6, Alternating, 5]][q, t]
Out[15]=	-3 1 1 1 3 3 1 2 1 q + - + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 6 11 5 11 4 9 4 9 3 7 2 5 2 q t q t q t q t q t q t q t 2 ---- 3 q t