L8a12

L8a11

L8a13

	Visit L8a12's page at Knotilus! Visit L8a12's page at the original Knot Atlas!
	L8a12 is $8_{2}^{2}$ in the Rolfsen table of links.

L8a12 Further Notes and Views

Knot presentations

Planar diagram presentation	X10,1,11,2 X16,7,9,8 X12,3,13,4 X14,5,15,6 X4,13,5,14 X6,15,7,16 X2,9,3,10 X8,11,1,12
Gauss code	{1, -7, 3, -5, 4, -6, 2, -8}, {7, -1, 8, -3, 5, -4, 6, -2}

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, $-{\frac {299}{12}}$)

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:L8a12/V 2,1 Data:L8a12/V 3,1 Data:L8a12/V 4,1 Data:L8a12/V 4,2 Data:L8a12/V 4,3 Data:L8a12/V 5,1 Data:L8a12/V 5,2 Data:L8a12/V 5,3 Data:L8a12/V 5,4 Data:L8a12/V 6,1 Data:L8a12/V 6,2 Data:L8a12/V 6,3 Data:L8a12/V 6,4 Data:L8a12/V 6,5 Data:L8a12/V 6,6 Data:L8a12/V 6,7 Data:L8a12/V 6,8 Data:L8a12/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 L8a12. 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             1     1 -10           1 1     0 -12         2 1       1 -14       1 1         0 -16     1 2           -1 -18     1             1 -20 1 1               0 -22 1                 1

Integral Khovanov Homology (db, data source)

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

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, 12]]
Out[2]=	8
In[3]:=	PD[Link[8, Alternating, 12]]
Out[3]=	PD[X[10, 1, 11, 2], X[16, 7, 9, 8], X[12, 3, 13, 4], X[14, 5, 15, 6], X[4, 13, 5, 14], X[6, 15, 7, 16], X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[4]:=	GaussCode[Link[8, Alternating, 12]]
Out[4]=	GaussCode[{1, -7, 3, -5, 4, -6, 2, -8}, {7, -1, 8, -3, 5, -4, 6, -2}]
In[5]:=	BR[Link[8, Alternating, 12]]
Out[5]=	BR[Link[8, Alternating, 12]]
In[6]:=	alex = Alexander[Link[8, Alternating, 12]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, Alternating, 12]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, Alternating, 12]], KnotSignature[Link[8, Alternating, 12]]}
Out[9]=	{Infinity, -5}
In[10]:=	J=Jones[Link[8, Alternating, 12]][q]
Out[10]=	-(21/2) -(19/2) 2 3 3 2 2 -(7/2) -q + q - ----- + ----- - ----- + ----- - ---- + q - 17/2 15/2 13/2 11/2 9/2 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, 12]][q]
Out[12]=	-32 -30 -28 -26 -22 -18 -16 -12 -8 q + q + q + q + q + q + q + q + q
In[13]:=	Kauffman[Link[8, Alternating, 12]][a, z]
Out[13]=	7 9 8 a a 5 7 9 11 13 6 2 a - -- - -- - 3 a z + 5 a z + 5 a z - a z + 2 a z - a z - z z 8 2 10 2 12 2 5 3 7 3 9 3 11 3 5 a z - 3 a z + a z + 4 a z - 6 a z - 8 a z + a z - 13 3 6 4 8 4 10 4 12 4 5 5 7 5 a z + 3 a z + 6 a z + 2 a z - a z - a z + 4 a z + 9 5 11 5 6 6 8 6 10 6 7 7 9 7 4 a z - a z - a z - 2 a z - a z - a z - a z
In[14]:=	{Vassiliev[2][Link[8, Alternating, 12]], Vassiliev[3][Link[8, Alternating, 12]]}
Out[14]=	299 {0, -(---)} 12
In[15]:=	Kh[Link[8, Alternating, 12]][q, t]
Out[15]=	-6 -4 1 1 1 1 1 2 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 22 8 20 8 20 7 18 6 16 6 16 5 q t q t q t q t q t q t 1 1 2 1 1 1 1 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