L8a10

L8a9

L8a11

	Visit L8a10's page at Knotilus! Visit L8a10's page at the original Knot Atlas!
	L8a10 is $8_{5}^{2}$ in the Rolfsen table of links.

Symmetric version

Mongolian ornament

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(2)t(1)^{2}-t(1)^{2}+2t(2)^{2}t(1)-3t(2)t(1)+2t(1)-t(2)^{2}+2t(2)}{t(1)t(2)}}$ (db)
Jones polynomial	$-{\frac {1}{q^{3/2}}}+{\frac {2}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {5}{q^{11/2}}}+{\frac {4}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z-a^{7}z^{3}+a^{7}z^{-1}-2a^{5}z^{3}-3a^{5}z-a^{5}z^{-1}-a^{3}z^{3}-a^{3}z$ (db)
Kauffman polynomial	$-z^{5}a^{11}+3z^{3}a^{11}-2za^{11}-2z^{6}a^{10}+6z^{4}a^{10}-4z^{2}a^{10}-z^{7}a^{9}+4z^{3}a^{9}-za^{9}-4z^{6}a^{8}+8z^{4}a^{8}-3z^{2}a^{8}-z^{7}a^{7}-2z^{5}a^{7}+6z^{3}a^{7}-4za^{7}+a^{7}z^{-1}-2z^{6}a^{6}+2z^{2}a^{6}-a^{6}-3z^{5}a^{5}+4z^{3}a^{5}-4za^{5}+a^{5}z^{-1}-2z^{4}a^{4}+z^{2}a^{4}-z^{3}a^{3}+za^{3}$ (db)

Vassiliev invariants

V2 and V3:

(0, $-{\frac {177}{16}}$)

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:L8a10/V 2,1 Data:L8a10/V 3,1 Data:L8a10/V 4,1 Data:L8a10/V 4,2 Data:L8a10/V 4,3 Data:L8a10/V 5,1 Data:L8a10/V 5,2 Data:L8a10/V 5,3 Data:L8a10/V 5,4 Data:L8a10/V 6,1 Data:L8a10/V 6,2 Data:L8a10/V 6,3 Data:L8a10/V 6,4 Data:L8a10/V 6,5 Data:L8a10/V 6,6 Data:L8a10/V 6,7 Data:L8a10/V 6,8 Data:L8a10/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 $s=$-3 is the signature of L8a10. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                 1 1 -4               2 1 -1 -6             2     2 -8           2 2     0 -10         3 2       1 -12       1 2         1 -14     2 3           -1 -16   1 2             1 -18   1               -1 -20 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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, 10]]
Out[2]=	8
In[3]:=	PD[Link[8, Alternating, 10]]
Out[3]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[12, 15, 13, 16], X[14, 5, 15, 6], X[4, 13, 5, 14], X[16, 11, 7, 12], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[4]:=	GaussCode[Link[8, Alternating, 10]]
Out[4]=	GaussCode[{1, -7, 2, -5, 4, -8}, {7, -1, 8, -2, 6, -3, 5, -4, 3, -6}]
In[5]:=	BR[Link[8, Alternating, 10]]
Out[5]=	BR[Link[8, Alternating, 10]]
In[6]:=	alex = Alexander[Link[8, Alternating, 10]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, Alternating, 10]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, Alternating, 10]], KnotSignature[Link[8, Alternating, 10]]}
Out[9]=	{Infinity, -3}
In[10]:=	J=Jones[Link[8, Alternating, 10]][q]
Out[10]=	-(19/2) 2 3 4 5 4 4 2 -(3/2) -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - q 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[8, Alternating, 10]][q]
Out[12]=	-30 -26 -24 -20 2 -16 2 -10 -8 -6 -4 q - q + q + q + --- + q + --- + q + q - q + q 18 14 q q
In[13]:=	Kauffman[Link[8, Alternating, 10]][a, z]
Out[13]=	5 7 6 a a 3 5 7 9 11 4 2 -a + -- + -- + a z - 4 a z - 4 a z - a z - 2 a z + a z + z z 6 2 8 2 10 2 3 3 5 3 7 3 9 3 2 a z - 3 a z - 4 a z - a z + 4 a z + 6 a z + 4 a z + 11 3 4 4 8 4 10 4 5 5 7 5 3 a z - 2 a z + 8 a z + 6 a z - 3 a z - 2 a z - 11 5 6 6 8 6 10 6 7 7 9 7 a z - 2 a z - 4 a z - 2 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[8, Alternating, 10]], Vassiliev[3][Link[8, Alternating, 10]]}
Out[14]=	177 {0, -(---)} 16
In[15]:=	Kh[Link[8, Alternating, 10]][q, t]
Out[15]=	-4 -2 1 1 1 2 2 3 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 20 8 18 7 16 7 16 6 14 6 14 5 q t q t q t q t q t q t 1 2 3 2 2 2 2 2 ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- 12 5 12 4 10 4 10 3 8 3 8 2 6 2 4 q t q t q t q t q t q t q t q t