L5a1

L4a1

L6a1

	Visit L5a1's page at Knotilus! Visit L5a1's page at the original Knot Atlas!
	L5a1 is [math]\displaystyle{ 5^2_1 }[/math] in Rolfsen's Table of Links. It is also known as the "Whitehead Link".

Basic depiction		Drawing of "Thor's hammer" or Mjölnir found in Sweden		Wolfgang Staubach's medallion based on this [1]
A kolam with two cycles, one of which is twisted[2]		A simplest closed-loop version of heraldic "fret" / "fretty" ornamentation.		Bisexuality symbol.

Knot presentations

Planar diagram presentation	X6172 X10,7,5,8 X4516 X2,10,3,9 X8493
Gauss code	{1, -4, 5, -3}, {3, -1, 2, -5, 4, -2}

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)}{\sqrt{u} \sqrt{v}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{1}{q^{7/2}}-\frac{2}{q^{5/2}}-q^{3/2}+\frac{1}{q^{3/2}}+\sqrt{q}-\frac{2}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^3+z^3 a+2 z a+a z^{-1} -z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^2 a^4-2 z^3 a^3+2 z a^3-z^4 a^2-3 z^3 a+4 z a-a z^{-1} -z^4+z^2+1-z^3 a^{-1} +2 z a^{-1} - a^{-1} z^{-1} }[/math] (db)

Vassiliev invariants

V2 and V3:

(0, [math]\displaystyle{ \frac{1}{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:L5a1/V 2,1 Data:L5a1/V 3,1 Data:L5a1/V 4,1 Data:L5a1/V 4,2 Data:L5a1/V 4,3 Data:L5a1/V 5,1 Data:L5a1/V 5,2 Data:L5a1/V 5,3 Data:L5a1/V 5,4 Data:L5a1/V 6,1 Data:L5a1/V 6,2 Data:L5a1/V 6,3 Data:L5a1/V 6,4 Data:L5a1/V 6,5 Data:L5a1/V 6,6 Data:L5a1/V 6,7 Data:L5a1/V 6,8 Data:L5a1/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]-1 is the signature of L5a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 χ 4           1 1 2             0 0       2 1   1 -2     1 2     1 -4   1         1 -6   1         1 -8 1           -1

Integral Khovanov Homology (db, data source)

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