L2a1

L2a1

L4a1

	Visit L2a1's page at Knotilus! Visit L2a1's page at the original Knot Atlas!
	L2a1 is $2_{1}^{2}$ in Rolfsen's table of links. It is also known as the "Hopf Link".

Japanese family emblem	Linked hearts	expanded Kolam Two-hearts [1]	Logo
In Star of David form on old Jewish building in Prague	Three wreaths linked as two L2a1 configurations on Michelangelo's tomb	One form of a heterosexuality symbol	(pseudo-3D)
Are they forever linked? [2]	As impossible object	Linked hearts (pseudo-3D)	German coat of arms

Knot presentations

Planar diagram presentation	X4132 X2314
Gauss code	{1, -2}, {2, -1}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-1$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{-1}-za-az^{-1}$ (db)
Kauffman polynomial	$-za^{3}+a^{3}z^{-1}-a^{2}-za+az^{-1}$ (db)

Vassiliev invariants

V2 and V3:

(0, $-{\frac {17}{48}}$)

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:L2a1/V 2,1 Data:L2a1/V 3,1 Data:L2a1/V 4,1 Data:L2a1/V 4,2 Data:L2a1/V 4,3 Data:L2a1/V 5,1 Data:L2a1/V 5,2 Data:L2a1/V 5,3 Data:L2a1/V 5,4 Data:L2a1/V 6,1 Data:L2a1/V 6,2 Data:L2a1/V 6,3 Data:L2a1/V 6,4 Data:L2a1/V 6,5 Data:L2a1/V 6,6 Data:L2a1/V 6,7 Data:L2a1/V 6,8 Data:L2a1/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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$-1 is the signature of L2a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-2$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-1$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Link[2, Alternating, 1]]
Out[2]=	2
In[3]:=	PD[Link[2, Alternating, 1]]
Out[3]=	PD[X[4, 1, 3, 2], X[2, 3, 1, 4]]
In[4]:=	GaussCode[Link[2, Alternating, 1]]
Out[4]=	GaussCode[{1, -2}, {2, -1}]
In[5]:=	BR[Link[2, Alternating, 1]]
Out[5]=	BR[Link[2, Alternating, 1]]
In[6]:=	alex = Alexander[Link[2, Alternating, 1]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[2, Alternating, 1]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[2, Alternating, 1]], KnotSignature[Link[2, Alternating, 1]]}
Out[9]=	{Infinity, -1}
In[10]:=	J=Jones[Link[2, Alternating, 1]][q]
Out[10]=	-(5/2) 1 -q - ------- Sqrt[q]
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[2, Alternating, 1]][q]
Out[12]=	-10 2 2 2 -2 1 + q + -- + -- + -- + q 8 6 4 q q q
In[13]:=	Kauffman[Link[2, Alternating, 1]][a, z]
Out[13]=	3 2 a a 3 -a + - + -- - a z - a z z z
In[14]:=	{Vassiliev[2][Link[2, Alternating, 1]], Vassiliev[3][Link[2, Alternating, 1]]}
Out[14]=	17 {0, -(--)} 48
In[15]:=	Kh[Link[2, Alternating, 1]][q, t]
Out[15]=	-2 1 1 1 + q + ----- + ----- 6 2 4 2 q t q t