L8a11

L8a10

L8a12

	Visit L8a11's page at Knotilus! Visit L8a11's page at the original Knot Atlas!
	L8a11 is $8_{3}^{2}$ in the Rolfsen table of links.

L8a11 Further Notes and Views

Knot presentations

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X16,11,7,12 X12,15,13,16 X6718 X4,13,5,14
Gauss code	{1, -2, 3, -8, 4, -7}, {7, -1, 2, -3, 5, -6, 8, -4, 6, -5}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(2)^{2}t(1)^{2}-t(2)t(1)^{2}-t(2)^{2}t(1)+3t(2)t(1)-t(1)-t(2)+2}{t(1)t(2)}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {4}{q^{15/2}}}-{\frac {3}{q^{17/2}}}+{\frac {2}{q^{19/2}}}-{\frac {1}{q^{21/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z^{3}a^{9}+2za^{9}-z^{5}a^{7}-3z^{3}a^{7}-za^{7}+a^{7}z^{-1}-z^{5}a^{5}-4z^{3}a^{5}-4za^{5}-a^{5}z^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{13}+za^{13}-2z^{4}a^{12}+2z^{2}a^{12}-2z^{5}a^{11}+z^{3}a^{11}-2z^{6}a^{10}+3z^{4}a^{10}-3z^{2}a^{10}-z^{7}a^{9}+z^{5}a^{9}-2z^{3}a^{9}+2za^{9}-3z^{6}a^{8}+7z^{4}a^{8}-4z^{2}a^{8}-z^{7}a^{7}+2z^{5}a^{7}-za^{7}+a^{7}z^{-1}-z^{6}a^{6}+2z^{4}a^{6}+z^{2}a^{6}-a^{6}-z^{5}a^{5}+4z^{3}a^{5}-4za^{5}+a^{5}z^{-1}$ (db)

Vassiliev invariants

V2 and V3:

(0, $-{\frac {193}{16}}$)

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:L8a11/V 2,1 Data:L8a11/V 3,1 Data:L8a11/V 4,1 Data:L8a11/V 4,2 Data:L8a11/V 4,3 Data:L8a11/V 5,1 Data:L8a11/V 5,2 Data:L8a11/V 5,3 Data:L8a11/V 5,4 Data:L8a11/V 6,1 Data:L8a11/V 6,2 Data:L8a11/V 6,3 Data:L8a11/V 6,4 Data:L8a11/V 6,5 Data:L8a11/V 6,6 Data:L8a11/V 6,7 Data:L8a11/V 6,8 Data:L8a11/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 $s=$-5 is the signature of L8a11. 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 -4                 1 1 -6               1 1 0 -8             2     2 -10           1 1     0 -12         3 2       1 -14       1 1         0 -16     2 3           -1 -18   1 2             1 -20   1               -1 -22 1                 1

Integral Khovanov Homology (db, data source)

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

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, 11]]
Out[2]=	8
In[3]:=	PD[Link[8, Alternating, 11]]
Out[3]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[14, 5, 15, 6], X[16, 11, 7, 12], X[12, 15, 13, 16], X[6, 7, 1, 8], X[4, 13, 5, 14]]
In[4]:=	GaussCode[Link[8, Alternating, 11]]
Out[4]=	GaussCode[{1, -2, 3, -8, 4, -7}, {7, -1, 2, -3, 5, -6, 8, -4, 6, -5}]
In[5]:=	BR[Link[8, Alternating, 11]]
Out[5]=	BR[Link[8, Alternating, 11]]
In[6]:=	alex = Alexander[Link[8, Alternating, 11]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, Alternating, 11]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, Alternating, 11]], KnotSignature[Link[8, Alternating, 11]]}
Out[9]=	{Infinity, -5}
In[10]:=	J=Jones[Link[8, Alternating, 11]][q]
Out[10]=	-(21/2) 2 3 4 4 3 3 -(7/2) -q + ----- - ----- + ----- - ----- + ----- - ---- + q - 19/2 17/2 15/2 13/2 11/2 9/2 q q q q q q -(5/2) q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[8, Alternating, 11]][q]
Out[12]=	-32 -24 -22 2 2 -14 2 -8 q - q + q + --- + --- + q + --- + q 18 16 12 q q q
In[13]:=	Kauffman[Link[8, Alternating, 11]][a, z]
Out[13]=	5 7 6 a a 5 7 9 13 6 2 8 2 -a + -- + -- - 4 a z - a z + 2 a z + a z + a z - 4 a z - z z 10 2 12 2 5 3 9 3 11 3 13 3 6 4 3 a z + 2 a z + 4 a z - 2 a z + a z - a z + 2 a z + 8 4 10 4 12 4 5 5 7 5 9 5 11 5 7 a z + 3 a z - 2 a z - a z + 2 a z + a z - 2 a z - 6 6 8 6 10 6 7 7 9 7 a z - 3 a z - 2 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[8, Alternating, 11]], Vassiliev[3][Link[8, Alternating, 11]]}
Out[14]=	193 {0, -(---)} 16
In[15]:=	Kh[Link[8, Alternating, 11]][q, t]
Out[15]=	-6 -4 1 1 1 2 2 3 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 22 8 20 7 18 7 18 6 16 6 16 5 q t q t q t q t q t q t 1 1 3 2 1 1 2 1 ------ + ------ + ------ + ------ + ------ + ------ + ----- + ---- 14 5 14 4 12 4 12 3 10 3 10 2 8 2 6 q t q t q t q t q t q t q t q t