L8a16

L8a15

L8a17

	Visit L8a16's page at Knotilus! Visit L8a16's page at the original Knot Atlas!
	L8a16 is $8_{5}^{3}$ in the Rolfsen table of links.

Depiction obtained by knotilus

Knot presentations

Planar diagram presentation	X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X16,10,11,9 X10,12,5,11 X4,15,1,16
Gauss code	{1, -4, 3, -8}, {2, -1, 5, -3, 6, -7}, {7, -2, 4, -5, 8, -6}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)(t(3)-1)(t(2)t(3)+1)}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{6}+3q^{5}-4q^{4}+6q^{3}-5q^{2}-q^{-2}+6q+3q^{-1}-3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+4z^{4}a^{-2}-z^{4}a^{-4}-z^{4}+4z^{2}a^{-2}-2z^{2}a^{-4}-2z^{2}-a^{-2}+1-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$2z^{7}a^{-1}+2z^{7}a^{-3}+7z^{6}a^{-2}+4z^{6}a^{-4}+3z^{6}+az^{5}-3z^{5}a^{-1}+4z^{5}a^{-5}-17z^{4}a^{-2}-5z^{4}a^{-4}+3z^{4}a^{-6}-9z^{4}-2az^{3}-z^{3}a^{-1}-3z^{3}a^{-3}-3z^{3}a^{-5}+z^{3}a^{-7}+10z^{2}a^{-2}+2z^{2}a^{-4}-2z^{2}a^{-6}+6z^{2}-za^{-1}-za^{-3}+a^{-2}+a^{-4}+1+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (db)

Vassiliev invariants

V2 and V3:

(0, $-{\frac {5}{6}}$)

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:L8a16/V 2,1 Data:L8a16/V 3,1 Data:L8a16/V 4,1 Data:L8a16/V 4,2 Data:L8a16/V 4,3 Data:L8a16/V 5,1 Data:L8a16/V 5,2 Data:L8a16/V 5,3 Data:L8a16/V 5,4 Data:L8a16/V 6,1 Data:L8a16/V 6,2 Data:L8a16/V 6,3 Data:L8a16/V 6,4 Data:L8a16/V 6,5 Data:L8a16/V 6,6 Data:L8a16/V 6,7 Data:L8a16/V 6,8 Data:L8a16/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=$2 is the signature of L8a16. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 χ 13                 1 -1 11               2   2 9             2 1   -1 7           4 2     2 5         3 4       1 3       3 2         1 1     2 5           3 -1   1 1             0 -3   2               2 -5 1                 -1

Integral Khovanov Homology (db, data source)

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

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, 16]]
Out[2]=	8
In[3]:=	PD[Link[8, Alternating, 16]]
Out[3]=	PD[X[6, 1, 7, 2], X[12, 6, 13, 5], X[8, 4, 9, 3], X[2, 14, 3, 13], X[14, 7, 15, 8], X[16, 10, 11, 9], X[10, 12, 5, 11], X[4, 15, 1, 16]]
In[4]:=	GaussCode[Link[8, Alternating, 16]]
Out[4]=	GaussCode[{1, -4, 3, -8}, {2, -1, 5, -3, 6, -7}, {7, -2, 4, -5, 8, -6}]
In[5]:=	BR[Link[8, Alternating, 16]]
Out[5]=	BR[Link[8, Alternating, 16]]
In[6]:=	alex = Alexander[Link[8, Alternating, 16]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, Alternating, 16]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, Alternating, 16]], KnotSignature[Link[8, Alternating, 16]]}
Out[9]=	{Infinity, 2}
In[10]:=	J=Jones[Link[8, Alternating, 16]][q]
Out[10]=	-2 3 2 3 4 5 6 -3 - q + - + 6 q - 5 q + 6 q - 4 q + 3 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[8, Alternating, 16]][q]
Out[12]=	-6 -4 -2 2 4 6 8 10 12 16 3 - q + q + q + 5 q + 3 q + 6 q + 3 q + 4 q + 2 q + q - 18 q
In[13]:=	Kauffman[Link[8, Alternating, 16]][a, z]
Out[13]=	-4 -2 -2 1 2 2 2 z z 2 1 + a + a - z - ----- - ----- + ---- + --- - -- - - + 6 z - 4 2 2 2 3 a z 3 a a z a z a z a 2 2 2 3 3 3 3 4 2 z 2 z 10 z z 3 z 3 z z 3 4 3 z ---- + ---- + ----- + -- - ---- - ---- - -- - 2 a z - 9 z + ---- - 6 4 2 7 5 3 a 6 a a a a a a a 4 4 5 5 6 6 7 7 5 z 17 z 4 z 3 z 5 6 4 z 7 z 2 z 2 z ---- - ----- + ---- - ---- + a z + 3 z + ---- + ---- + ---- + ---- 4 2 5 a 4 2 3 a a a a a a a
In[14]:=	{Vassiliev[2][Link[8, Alternating, 16]], Vassiliev[3][Link[8, Alternating, 16]]}
Out[14]=	5 {0, -(-)} 6
In[15]:=	Kh[Link[8, Alternating, 16]][q, t]
Out[15]=	3 1 2 1 1 2 q 3 5 5 q + 3 q + ----- + ----- + ---- + --- + --- + 2 q t + 3 q t + 5 3 3 2 2 q t t q t q t q t 5 2 7 2 7 3 9 3 9 4 11 4 13 5 4 q t + 4 q t + 2 q t + 2 q t + q t + 2 q t + q t