L8n3

L8n2

L8n4

	Visit L8n3's page at Knotilus! Visit L8n3's page at the original Knot Atlas!
	L8n3 is $8_{7}^{3}$ in the Rolfsen table of links.

L8n3 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X5,12,6,13 X3849 X13,2,14,3 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 $u$, $v$, $w$, ...)	${\frac {uv^{2}w^{2}-1}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$q^{-3}+q^{-5}+q^{-7}+q^{-9}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a^{10}z^{-2}+a^{10}-z^{4}a^{8}-5z^{2}a^{8}-2a^{8}z^{-2}-6a^{8}+z^{6}a^{6}+6z^{4}a^{6}+10z^{2}a^{6}+a^{6}z^{-2}+5a^{6}$ (db)
Kauffman polynomial	$a^{12}+a^{10}z^{2}+a^{10}z^{-2}-3a^{10}+a^{9}z^{5}-5a^{9}z^{3}+6a^{9}z-2a^{9}z^{-1}+a^{8}z^{6}-6a^{8}z^{4}+11a^{8}z^{2}+2a^{8}z^{-2}-8a^{8}+a^{7}z^{5}-5a^{7}z^{3}+6a^{7}z-2a^{7}z^{-1}+a^{6}z^{6}-6a^{6}z^{4}+10a^{6}z^{2}+a^{6}z^{-2}-5a^{6}$ (db)

Vassiliev invariants

V2 and V3:

(0, ${\frac {100}{3}}$)

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:L8n3/V 2,1 Data:L8n3/V 3,1 Data:L8n3/V 4,1 Data:L8n3/V 4,2 Data:L8n3/V 4,3 Data:L8n3/V 5,1 Data:L8n3/V 5,2 Data:L8n3/V 5,3 Data:L8n3/V 5,4 Data:L8n3/V 6,1 Data:L8n3/V 6,2 Data:L8n3/V 6,3 Data:L8n3/V 6,4 Data:L8n3/V 6,5 Data:L8n3/V 6,6 Data:L8n3/V 6,7 Data:L8n3/V 6,8 Data:L8n3/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 $s=$-6 is the signature of L8n3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 χ -5             1 1 -7             1 1 -9         1     1 -11     1         1 -13     2 1       1 -15 1             1 -17 2 1           1 -19 1             1

Integral Khovanov Homology (db, data source)

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

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, NonAlternating, 3]]
Out[2]=	8
In[3]:=	PD[Link[8, NonAlternating, 3]]
Out[3]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[13, 2, 14, 3], 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, 3]]
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, 3]]
Out[5]=	BR[Link[8, NonAlternating, 3]]
In[6]:=	alex = Alexander[Link[8, NonAlternating, 3]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, NonAlternating, 3]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, NonAlternating, 3]], KnotSignature[Link[8, NonAlternating, 3]]}
Out[9]=	{Infinity, -6}
In[10]:=	J=Jones[Link[8, NonAlternating, 3]][q]
Out[10]=	-9 -7 -5 -3 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, 3]][q]
Out[12]=	-32 -30 2 3 4 4 3 3 2 2 -12 q + q + --- + --- + --- + --- + --- + --- + --- + --- + q + 28 26 24 22 20 18 16 14 q q q q q q q q -10 q
In[13]:=	Kauffman[Link[8, NonAlternating, 3]][a, z]
Out[13]=	6 8 10 7 9 6 8 10 12 a 2 a a 2 a 2 a 7 -5 a - 8 a - 3 a + a + -- + ---- + --- - ---- - ---- + 6 a z + 2 2 2 z z z z z 9 6 2 8 2 10 2 7 3 9 3 6 4 6 a z + 10 a z + 11 a z + a z - 5 a z - 5 a z - 6 a z - 8 4 7 5 9 5 6 6 8 6 6 a z + a z + a z + a z + a z
In[14]:=	{Vassiliev[2][Link[8, NonAlternating, 3]], Vassiliev[3][Link[8, NonAlternating, 3]]}
Out[14]=	100 {0, ---} 3
In[15]:=	Kh[Link[8, NonAlternating, 3]][q, t]
Out[15]=	-7 -5 1 2 1 1 2 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 19 6 17 6 15 6 17 5 13 4 11 4 q t q t q t q t q t q t 1 1 ------ + ----- 13 3 9 2 q t q t