L8a18

L8a17

L8a19

	Visit L8a18's page at Knotilus! Visit L8a18's page at the original Knot Atlas!
	L8a18 is $8_{1}^{3}$ in the Rolfsen table of links.

L8a18 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X12,4,13,3 X8,16,9,15 X14,8,15,7 X16,10,11,9 X10,12,5,11 X2536 X4,14,1,13
Gauss code	{1, -7, 2, -8}, {7, -1, 4, -3, 5, -6}, {6, -2, 8, -4, 3, -5}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{2}w^{2}-uvw^{2}+uvw-uw+u-v^{2}w^{2}+v^{2}w-vw+v-1}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$q^{8}-2q^{7}+3q^{6}-3q^{5}+4q^{4}-2q^{3}+3q^{2}-q+1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{4}a^{-6}+3z^{2}a^{-6}+a^{-6}z^{-2}+2a^{-6}-z^{6}a^{-4}-5z^{4}a^{-4}-8z^{2}a^{-4}-2a^{-4}z^{-2}-6a^{-4}+z^{4}a^{-2}+4z^{2}a^{-2}+a^{-2}z^{-2}+4a^{-2}$ (db)
Kauffman polynomial	$z^{2}a^{-10}+2z^{3}a^{-9}+3z^{4}a^{-8}-3z^{2}a^{-8}+a^{-8}+3z^{5}a^{-7}-4z^{3}a^{-7}+3z^{6}a^{-6}-8z^{4}a^{-6}+6z^{2}a^{-6}+a^{-6}z^{-2}-3a^{-6}+z^{7}a^{-5}-7z^{3}a^{-5}+6za^{-5}-2a^{-5}z^{-1}+4z^{6}a^{-4}-16z^{4}a^{-4}+18z^{2}a^{-4}+2a^{-4}z^{-2}-8a^{-4}+z^{7}a^{-3}-3z^{5}a^{-3}-z^{3}a^{-3}+6za^{-3}-2a^{-3}z^{-1}+z^{6}a^{-2}-5z^{4}a^{-2}+8z^{2}a^{-2}+a^{-2}z^{-2}-5a^{-2}$ (db)

Vassiliev invariants

V2 and V3:

(0, $-{\frac {26}{3}}$)

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:L8a18/V 2,1 Data:L8a18/V 3,1 Data:L8a18/V 4,1 Data:L8a18/V 4,2 Data:L8a18/V 4,3 Data:L8a18/V 5,1 Data:L8a18/V 5,2 Data:L8a18/V 5,3 Data:L8a18/V 5,4 Data:L8a18/V 6,1 Data:L8a18/V 6,2 Data:L8a18/V 6,3 Data:L8a18/V 6,4 Data:L8a18/V 6,5 Data:L8a18/V 6,6 Data:L8a18/V 6,7 Data:L8a18/V 6,8 Data:L8a18/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=$4 is the signature of L8a18. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 χ 17                 1 1 15               2 1 -1 13             1     1 11           2 2     0 9         2 1       1 7       1 3         2 5     2 1           1 3   1 3             2 1                   0 -1 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=5$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=6$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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, Alternating, 18]]
Out[2]=	8
In[3]:=	PD[Link[8, Alternating, 18]]
Out[3]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[8, 16, 9, 15], X[14, 8, 15, 7], X[16, 10, 11, 9], X[10, 12, 5, 11], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[4]:=	GaussCode[Link[8, Alternating, 18]]
Out[4]=	GaussCode[{1, -7, 2, -8}, {7, -1, 4, -3, 5, -6}, {6, -2, 8, -4, 3, -5}]
In[5]:=	BR[Link[8, Alternating, 18]]
Out[5]=	BR[Link[8, Alternating, 18]]
In[6]:=	alex = Alexander[Link[8, Alternating, 18]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, Alternating, 18]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, Alternating, 18]], KnotSignature[Link[8, Alternating, 18]]}
Out[9]=	{Infinity, 4}
In[10]:=	J=Jones[Link[8, Alternating, 18]][q]
Out[10]=	2 3 4 5 6 7 8 1 - q + 3 q - 2 q + 4 q - 3 q + 3 q - 2 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[8, Alternating, 18]][q]
Out[12]=	2 4 6 8 10 12 14 16 18 1 + q + 2 q + 3 q + 3 q + 5 q + 3 q + 4 q + 2 q + q + 20 24 q + q
In[13]:=	Kauffman[Link[8, Alternating, 18]][a, z]
Out[13]=	-8 3 8 5 1 2 1 2 2 6 z 6 z a - -- - -- - -- + ----- + ----- + ----- - ---- - ---- + --- + --- + 6 4 2 6 2 4 2 2 2 5 3 5 3 a a a a z a z a z a z a z a a 2 2 2 2 2 3 3 3 3 4 z 3 z 6 z 18 z 8 z 2 z 4 z 7 z z 3 z --- - ---- + ---- + ----- + ---- + ---- - ---- - ---- - -- + ---- - 10 8 6 4 2 9 7 5 3 8 a a a a a a a a a a 4 4 4 5 5 6 6 6 7 7 8 z 16 z 5 z 3 z 3 z 3 z 4 z z z z ---- - ----- - ---- + ---- - ---- + ---- + ---- + -- + -- + -- 6 4 2 7 3 6 4 2 5 3 a a a a a a a a a a
In[14]:=	{Vassiliev[2][Link[8, Alternating, 18]], Vassiliev[3][Link[8, Alternating, 18]]}
Out[14]=	26 {0, -(--)} 3
In[15]:=	Kh[Link[8, Alternating, 18]][q, t]
Out[15]=	3 3 5 1 q 5 7 7 2 9 2 9 3 3 q + 2 q + ---- + -- + q t + q t + 3 q t + 2 q t + q t + 2 t q t 11 3 11 4 13 4 15 5 15 6 17 6 2 q t + 2 q t + q t + 2 q t + q t + q t