L8a8

L8a7

L8a9

	Visit L8a8's page at Knotilus! Visit L8a8's page at the original Knot Atlas!
	L8a8 is $8_{7}^{2}$ in the Rolfsen table of links, and the "seized Carrick bend" of practical knot-tying.

The simplest Celtic or pseudo-Celtic linear decorative knot.		Alternate decorative variant		Circular arcs only		Decorative variant with big loops at ends
Coat of arms of Bressauc, Jura, Switzerland.

Knot presentations

Planar diagram presentation	X8192 X10,4,11,3 X16,10,7,9 X2738 X14,12,15,11 X12,5,13,6 X4,13,5,14 X6,16,1,15
Gauss code	{1, -4, 2, -7, 6, -8}, {4, -1, 3, -2, 5, -6, 7, -5, 8, -3}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{2}-2u^{2}v+u^{2}-2uv^{2}+3uv-2u+v^{2}-2v+1}{uv}}$ (db)
Jones polynomial	$-q^{9/2}+3q^{7/2}-4q^{5/2}+5q^{3/2}-6{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {2}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-3}+a^{3}z-za^{-3}+a^{3}z^{-1}+z^{5}a^{-1}-2az^{3}+3z^{3}a^{-1}-4az+3za^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-az^{7}-z^{7}a^{-1}-2a^{2}z^{6}-3z^{6}a^{-2}-5z^{6}-a^{3}z^{5}-2az^{5}-5z^{5}a^{-1}-4z^{5}a^{-3}+5a^{2}z^{4}+z^{4}a^{-2}-3z^{4}a^{-4}+9z^{4}+3a^{3}z^{3}+10az^{3}+12z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-2a^{2}z^{2}+2z^{2}a^{-2}+2z^{2}a^{-4}-2z^{2}-3a^{3}z-7az-6za^{-1}-2za^{-3}-a^{2}+a^{3}z^{-1}+az^{-1}$ (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:L8a8/V 2,1 Data:L8a8/V 3,1 Data:L8a8/V 4,1 Data:L8a8/V 4,2 Data:L8a8/V 4,3 Data:L8a8/V 5,1 Data:L8a8/V 5,2 Data:L8a8/V 5,3 Data:L8a8/V 5,4 Data:L8a8/V 6,1 Data:L8a8/V 6,2 Data:L8a8/V 6,3 Data:L8a8/V 6,4 Data:L8a8/V 6,5 Data:L8a8/V 6,6 Data:L8a8/V 6,7 Data:L8a8/V 6,8 Data:L8a8/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 L8a8. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 χ 10                 1 1 8               2   -2 6             2 1   1 4           3 2     -1 2         3 2       1 0       2 4         2 -2     2 2           0 -4   1 3             2 -6   1               -1 -8 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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[8, Alternating, 8]]
Out[2]=	8
In[3]:=	PD[Link[8, Alternating, 8]]
Out[3]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[16, 10, 7, 9], X[2, 7, 3, 8], X[14, 12, 15, 11], X[12, 5, 13, 6], X[4, 13, 5, 14], X[6, 16, 1, 15]]
In[4]:=	GaussCode[Link[8, Alternating, 8]]
Out[4]=	GaussCode[{1, -4, 2, -7, 6, -8}, {4, -1, 3, -2, 5, -6, 7, -5, 8, -3}]
In[5]:=	BR[Link[8, Alternating, 8]]
Out[5]=	BR[Link[8, Alternating, 8]]
In[6]:=	alex = Alexander[Link[8, Alternating, 8]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, Alternating, 8]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, Alternating, 8]], KnotSignature[Link[8, Alternating, 8]]}
Out[9]=	{Infinity, 1}
In[10]:=	J=Jones[Link[8, Alternating, 8]][q]
Out[10]=	-(7/2) 2 4 4 3/2 5/2 -q + ---- - ---- + ------- - 6 Sqrt[q] + 5 q - 4 q + 5/2 3/2 Sqrt[q] q q 7/2 9/2 3 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[8, Alternating, 8]][q]
Out[12]=	-12 -10 2 -4 2 4 6 12 14 3 + q + q + -- + q + -- + q - 2 q - q + q 6 2 q q
In[13]:=	Kauffman[Link[8, Alternating, 8]][a, z]
Out[13]=	3 2 2 2 a a 2 z 6 z 3 2 2 z 2 z -a + - + -- - --- - --- - 7 a z - 3 a z - 2 z + ---- + ---- - z z 3 a 4 2 a a a 3 3 3 4 4 2 2 z 4 z 12 z 3 3 3 4 3 z z 2 a z - -- + ---- + ----- + 10 a z + 3 a z + 9 z - ---- + -- + 5 3 a 4 2 a a a a 5 5 6 7 2 4 4 z 5 z 5 3 5 6 3 z 2 6 z 5 a z - ---- - ---- - 2 a z - a z - 5 z - ---- - 2 a z - -- - 3 a 2 a a a 7 a z
In[14]:=	{Vassiliev[2][Link[8, Alternating, 8]], Vassiliev[3][Link[8, Alternating, 8]]}
Out[14]=	17 {0, -(--)} 48
In[15]:=	Kh[Link[8, Alternating, 8]][q, t]
Out[15]=	2 1 1 1 3 2 2 2 2 4 + 3 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 2 q t + 8 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 4 2 6 2 6 3 8 3 10 4 3 q t + 2 q t + 2 q t + q t + 2 q t + q t