L8n5

L8n4

L8n6

	Visit L8n5's page at Knotilus! Visit L8n5's page at the original Knot Atlas!
	L8n5 is is $8_{9}^{3}$ in the Rolfsen table of links.

L8n5 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X9,16,10,11 X11,10,12,5 X4,15,1,16
Gauss code	{1, -4, 3, -8}, {-2, -1, 5, -3, -6, 7}, {-7, 2, 4, -5, 8, 6}

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, ${\frac {20}{3}}$)

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:L8n5/V 2,1 Data:L8n5/V 3,1 Data:L8n5/V 4,1 Data:L8n5/V 4,2 Data:L8n5/V 4,3 Data:L8n5/V 5,1 Data:L8n5/V 5,2 Data:L8n5/V 5,3 Data:L8n5/V 5,4 Data:L8n5/V 6,1 Data:L8n5/V 6,2 Data:L8n5/V 6,3 Data:L8n5/V 6,4 Data:L8n5/V 6,5 Data:L8n5/V 6,6 Data:L8n5/V 6,7 Data:L8n5/V 6,8 Data:L8n5/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 $s=$-2 is the signature of L8n5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 χ -1             2 2 -3           2 2 0 -5         2     2 -7         2     2 -9     3 2       1 -11   1 2         1 -13   1           -1 -15 1             1

Integral Khovanov Homology (db, data source)

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

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