L8a3

L8a2

L8a4

	Visit L8a3's page at Knotilus! Visit L8a3's page at the original Knot Atlas!
	L8a3 is $8_{9}^{2}$ in the Rolfsen table of links.

L8a3 Further Notes and Views

Knot presentations

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

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, $-{\frac {125}{24}}$)

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:L8a3/V 2,1 Data:L8a3/V 3,1 Data:L8a3/V 4,1 Data:L8a3/V 4,2 Data:L8a3/V 4,3 Data:L8a3/V 5,1 Data:L8a3/V 5,2 Data:L8a3/V 5,3 Data:L8a3/V 5,4 Data:L8a3/V 6,1 Data:L8a3/V 6,2 Data:L8a3/V 6,3 Data:L8a3/V 6,4 Data:L8a3/V 6,5 Data:L8a3/V 6,6 Data:L8a3/V 6,7 Data:L8a3/V 6,8 Data:L8a3/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 $s=$-1 is the signature of L8a3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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, 3]]
Out[2]=	8
In[3]:=	PD[Link[8, Alternating, 3]]
Out[3]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 8, 15, 7], X[16, 11, 5, 12], X[12, 15, 13, 16], X[8, 14, 9, 13], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[4]:=	GaussCode[Link[8, Alternating, 3]]
Out[4]=	GaussCode[{1, -7, 2, -8}, {7, -1, 3, -6, 8, -2, 4, -5, 6, -3, 5, -4}]
In[5]:=	BR[Link[8, Alternating, 3]]
Out[5]=	BR[Link[8, Alternating, 3]]
In[6]:=	alex = Alexander[Link[8, Alternating, 3]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, Alternating, 3]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, Alternating, 3]], KnotSignature[Link[8, Alternating, 3]]}
Out[9]=	{Infinity, -1}
In[10]:=	J=Jones[Link[8, Alternating, 3]][q]
Out[10]=	-(13/2) 2 4 4 5 5 4 -q + ----- - ---- + ---- - ---- + ---- - ------- + 2 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[8, Alternating, 3]][q]
Out[12]=	-22 2 -16 -14 -12 -10 -6 -4 2 6 2 + q + --- + q + q - q + q + q + q - q + q 20 q
In[13]:=	Kauffman[Link[8, Alternating, 3]][a, z]
Out[13]=	3 5 7 4 a 2 a 2 a a z 3 5 7 2 a + - + ---- + ---- + -- + - - 3 a z - 9 a z - 8 a z - 3 a z + z - z z z z a 3 2 2 4 2 6 2 z 3 3 3 5 3 3 a z - 6 a z - 2 a z - -- + 3 a z + 12 a z + 11 a z + a 7 3 4 2 4 4 4 6 4 5 3 5 3 a z - 2 z + 4 a z + 11 a z + 5 a z - 3 a z - 4 a z - 5 5 7 5 2 6 4 6 6 6 3 7 5 7 2 a z - a z - 3 a z - 5 a z - 2 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[8, Alternating, 3]], Vassiliev[3][Link[8, Alternating, 3]]}
Out[14]=	125 {0, -(---)} 24
In[15]:=	Kh[Link[8, Alternating, 3]][q, t]
Out[15]=	2 1 1 1 3 2 2 2 3 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 q q t q t q t q t q t q t q t 3 2 2 3 2 4 2 ----- + ----- + ---- + ---- + t + q t + q t 6 2 4 2 4 2 q t q t q t q t