L8a19

L8a18

L8a20

	Visit L8a19's page at Knotilus! Visit L8a19's page at the original Knot Atlas!
	L8a19 is $8_{6}^{3}$ in the Rolfsen table of links.

L8a19 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X12,4,13,3 X14,5,15,6 X8,12,9,11 X16,8,11,7 X10,13,5,14 X2,9,3,10 X4,16,1,15
Gauss code	{1, -7, 2, -8}, {3, -1, 5, -4, 7, -6}, {4, -2, 6, -3, 8, -5}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(3)t(2)-t(2)+1)(t(2)t(3)-t(3)+1)}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{4}+q^{-4}-3q^{3}-3q^{-3}+5q^{2}+5q^{-2}-5q-5q^{-1}+8$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}+a^{2}z^{4}+z^{4}a^{-2}-4z^{4}+2a^{2}z^{2}+2z^{2}a^{-2}-5z^{2}+a^{2}+a^{-2}-2+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$2az^{7}+2z^{7}a^{-1}+4a^{2}z^{6}+4z^{6}a^{-2}+8z^{6}+3a^{3}z^{5}+az^{5}+z^{5}a^{-1}+3z^{5}a^{-3}+a^{4}z^{4}-7a^{2}z^{4}-7z^{4}a^{-2}+z^{4}a^{-4}-16z^{4}-4a^{3}z^{3}-4az^{3}-4z^{3}a^{-1}-4z^{3}a^{-3}-a^{4}z^{2}+5a^{2}z^{2}+5z^{2}a^{-2}-z^{2}a^{-4}+12z^{2}+2az+2za^{-1}-2a^{2}-2a^{-2}-3-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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:L8a19/V 2,1 Data:L8a19/V 3,1 Data:L8a19/V 4,1 Data:L8a19/V 4,2 Data:L8a19/V 4,3 Data:L8a19/V 5,1 Data:L8a19/V 5,2 Data:L8a19/V 5,3 Data:L8a19/V 5,4 Data:L8a19/V 6,1 Data:L8a19/V 6,2 Data:L8a19/V 6,3 Data:L8a19/V 6,4 Data:L8a19/V 6,5 Data:L8a19/V 6,6 Data:L8a19/V 6,7 Data:L8a19/V 6,8 Data:L8a19/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 L8a19. 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 9                 1 1 7               2   -2 5             3 1   2 3           3 3     0 1         5 2       3 -1       2 5         3 -3     3 3           0 -5   1 3             2 -7   2               -2 -9 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{5}$ $r=1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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, 19]]
Out[2]=	8
In[3]:=	PD[Link[8, Alternating, 19]]
Out[3]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 5, 15, 6], X[8, 12, 9, 11], X[16, 8, 11, 7], X[10, 13, 5, 14], X[2, 9, 3, 10], X[4, 16, 1, 15]]
In[4]:=	GaussCode[Link[8, Alternating, 19]]
Out[4]=	GaussCode[{1, -7, 2, -8}, {3, -1, 5, -4, 7, -6}, {4, -2, 6, -3, 8, -5}]
In[5]:=	BR[Link[8, Alternating, 19]]
Out[5]=	BR[Link[8, Alternating, 19]]
In[6]:=	alex = Alexander[Link[8, Alternating, 19]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, Alternating, 19]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, Alternating, 19]], KnotSignature[Link[8, Alternating, 19]]}
Out[9]=	{Infinity, 0}
In[10]:=	J=Jones[Link[8, Alternating, 19]][q]
Out[10]=	-4 3 5 5 2 3 4 8 + q - -- + -- - - - 5 q + 5 q - 3 q + q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[8, Alternating, 19]][q]
Out[12]=	-12 -10 -8 -6 2 7 2 4 6 8 10 5 + q - q + q + q + -- + -- + 7 q + 2 q + q + q - q + 4 2 q q 12 q
In[13]:=	Kauffman[Link[8, Alternating, 19]][a, z]
Out[13]=	2 2 2 2 1 a 2 2 a 2 z 2 -3 - -- - 2 a + -- + ----- + -- - --- - --- + --- + 2 a z + 12 z - 2 2 2 2 2 a z z a a z a z z 2 2 3 3 z 5 z 2 2 4 2 4 z 4 z 3 3 3 -- + ---- + 5 a z - a z - ---- - ---- - 4 a z - 4 a z - 4 2 3 a a a a 4 4 5 5 4 z 7 z 2 4 4 4 3 z z 5 3 5 16 z + -- - ---- - 7 a z + a z + ---- + -- + a z + 3 a z + 4 2 3 a a a a 6 7 6 4 z 2 6 2 z 7 8 z + ---- + 4 a z + ---- + 2 a z 2 a a
In[14]:=	{Vassiliev[2][Link[8, Alternating, 19]], Vassiliev[3][Link[8, Alternating, 19]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Link[8, Alternating, 19]][q, t]
Out[15]=	5 1 2 1 3 3 3 2 - + 5 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 2 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 9 4 3 q t + 3 q t + 3 q t + q t + 2 q t + q t