L8a5

L8a4

L8a6

	Visit L8a5's page at Knotilus! Visit L8a5's page at the original Knot Atlas!
	L8a5 is $8_{11}^{2}$ in the Rolfsen table of links.

L8a5 Further Notes and Views

Knot presentations

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

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, $-{\frac {17}{24}}$)

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:L8a5/V 2,1 Data:L8a5/V 3,1 Data:L8a5/V 4,1 Data:L8a5/V 4,2 Data:L8a5/V 4,3 Data:L8a5/V 5,1 Data:L8a5/V 5,2 Data:L8a5/V 5,3 Data:L8a5/V 5,4 Data:L8a5/V 6,1 Data:L8a5/V 6,2 Data:L8a5/V 6,3 Data:L8a5/V 6,4 Data:L8a5/V 6,5 Data:L8a5/V 6,6 Data:L8a5/V 6,7 Data:L8a5/V 6,8 Data:L8a5/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 L8a5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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