L8a1

L7n2

L8a2

	Visit L8a1's page at Knotilus! Visit L8a1's page at the original Knot Atlas!
	L8a1 is $8_{13}^{2}$ in the Rolfsen table of links.

L8a1 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X12,10,13,9 X8493 X10,5,11,6 X16,11,5,12 X2,14,3,13
Gauss code	{1, -8, 5, -3}, {6, -1, 2, -5, 4, -6, 7, -4, 8, -2, 3, -7}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}-3t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+5{\sqrt {q}}-{\frac {7}{\sqrt {q}}}+{\frac {7}{q^{3/2}}}-{\frac {7}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}(-z)+2a^{3}z^{3}+2a^{3}z-az^{5}-2az^{3}+z^{3}a^{-1}-az+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}-a^{6}z^{2}+3a^{5}z^{5}-4a^{5}z^{3}+2a^{5}z+4a^{4}z^{6}-5a^{4}z^{4}+2a^{4}z^{2}+2a^{3}z^{7}+4a^{3}z^{5}-10a^{3}z^{3}+4a^{3}z+9a^{2}z^{6}-14a^{2}z^{4}+z^{4}a^{-2}+5a^{2}z^{2}+2az^{7}+5az^{5}+4z^{5}a^{-1}-11az^{3}-5z^{3}a^{-1}+2az+az^{-1}+a^{-1}z^{-1}+5z^{6}-7z^{4}+2z^{2}-1$ (db)

Vassiliev invariants

V2 and V3:

(0, ${\frac {1}{2}}$)

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:L8a1/V 2,1 Data:L8a1/V 3,1 Data:L8a1/V 4,1 Data:L8a1/V 4,2 Data:L8a1/V 4,3 Data:L8a1/V 5,1 Data:L8a1/V 5,2 Data:L8a1/V 5,3 Data:L8a1/V 5,4 Data:L8a1/V 6,1 Data:L8a1/V 6,2 Data:L8a1/V 6,3 Data:L8a1/V 6,4 Data:L8a1/V 6,5 Data:L8a1/V 6,6 Data:L8a1/V 6,7 Data:L8a1/V 6,8 Data:L8a1/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 L8a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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