L8n8

L8n7

L9a1

	Visit L8n8's page at Knotilus! Visit L8n8's page at the original Knot Atlas!
	L8n8 is $8_{3}^{4}$ in the Rolfsen table of links.

Detail from an 18th century royal decree, Vietnam.

Knot presentations

Planar diagram presentation	X6172 X2536 X16,11,13,12 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X12,15,9,16
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, 3, -8}, {-7, 6, 8, -3}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(2)-t(3))(t(1)-t(4))}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}{\sqrt {t(4)}}}}$ (db)
Jones polynomial	$-q^{7/2}-q^{3/2}-2{\sqrt {q}}-{\frac {2}{\sqrt {q}}}-{\frac {1}{q^{3/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{3}z^{-3}-a^{-3}z^{-3}+a^{3}z+2a^{3}z^{-1}-za^{-3}-2a^{-3}z^{-1}-az^{3}-3az^{-3}+z^{3}a^{-1}+3a^{-1}z^{-3}-5az-6az^{-1}+5za^{-1}+6a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{5}-az^{5}-z^{5}a^{-1}-z^{5}a^{-3}-a^{2}z^{4}-z^{4}a^{-2}-2z^{4}+5a^{3}z^{3}+7az^{3}+7z^{3}a^{-1}+5z^{3}a^{-3}+6a^{2}z^{2}+6z^{2}a^{-2}+12z^{2}-6a^{3}z-14az-14za^{-1}-6za^{-3}-8a^{2}-8a^{-2}-15+4a^{3}z^{-1}+9az^{-1}+9a^{-1}z^{-1}+4a^{-3}z^{-1}+3a^{2}z^{-2}+3a^{-2}z^{-2}+6z^{-2}-a^{3}z^{-3}-3az^{-3}-3a^{-1}z^{-3}-a^{-3}z^{-3}$ (db)

Vassiliev invariants

V2 and V3:

(0, 0)

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:L8n8/V 2,1 Data:L8n8/V 3,1 Data:L8n8/V 4,1 Data:L8n8/V 4,2 Data:L8n8/V 4,3 Data:L8n8/V 5,1 Data:L8n8/V 5,2 Data:L8n8/V 5,3 Data:L8n8/V 5,4 Data:L8n8/V 6,1 Data:L8n8/V 6,2 Data:L8n8/V 6,3 Data:L8n8/V 6,4 Data:L8n8/V 6,5 Data:L8n8/V 6,6 Data:L8n8/V 6,7 Data:L8n8/V 6,8 Data:L8n8/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=$0 is the signature of L8n8. 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 8                 1 1 6                 1 1 4             1     1 2         3         3 0       1 6 1       4 -2         3         3 -4     1             1 -6 1                 1 -8 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $i=2$ $r=-4$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-3$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }^{6}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=3$ $r=4$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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