L8a17

L8a16

L8a18

	Visit L8a17's page at Knotilus! Visit L8a17's page at the original Knot Atlas!
	L8a17 is $8_{2}^{3}$ in the Rolfsen table of links.

L8a17 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(3)^{2}t(2)^{2}+t(1)t(3)t(2)^{2}-t(3)t(2)^{2}+t(1)t(3)^{2}t(2)-t(3)^{2}t(2)+t(1)t(2)-2t(1)t(3)t(2)+2t(3)t(2)-t(2)-t(1)+t(1)t(3)-t(3)}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{-10}-2q^{-9}+4q^{-8}-4q^{-7}+6q^{-6}-4q^{-5}+4q^{-4}-2q^{-3}+q^{-2}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{10}z^{-2}+a^{10}-3a^{8}z^{2}-2a^{8}z^{-2}-6a^{8}+2a^{6}z^{4}+6a^{6}z^{2}+a^{6}z^{-2}+5a^{6}+a^{4}z^{4}+2a^{4}z^{2}$ (db)
Kauffman polynomial	$a^{12}z^{4}-2a^{12}z^{2}+a^{12}+2a^{11}z^{5}-3a^{11}z^{3}+2a^{10}z^{6}-2a^{10}z^{4}+a^{10}z^{2}+a^{10}z^{-2}-3a^{10}+a^{9}z^{7}+2a^{9}z^{5}-6a^{9}z^{3}+6a^{9}z-2a^{9}z^{-1}+5a^{8}z^{6}-12a^{8}z^{4}+15a^{8}z^{2}+2a^{8}z^{-2}-8a^{8}+a^{7}z^{7}+2a^{7}z^{5}-6a^{7}z^{3}+6a^{7}z-2a^{7}z^{-1}+3a^{6}z^{6}-8a^{6}z^{4}+10a^{6}z^{2}+a^{6}z^{-2}-5a^{6}+2a^{5}z^{5}-3a^{5}z^{3}+a^{4}z^{4}-2a^{4}z^{2}$ (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:L8a17/V 2,1 Data:L8a17/V 3,1 Data:L8a17/V 4,1 Data:L8a17/V 4,2 Data:L8a17/V 4,3 Data:L8a17/V 5,1 Data:L8a17/V 5,2 Data:L8a17/V 5,3 Data:L8a17/V 5,4 Data:L8a17/V 6,1 Data:L8a17/V 6,2 Data:L8a17/V 6,3 Data:L8a17/V 6,4 Data:L8a17/V 6,5 Data:L8a17/V 6,6 Data:L8a17/V 6,7 Data:L8a17/V 6,8 Data:L8a17/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 $s=$-4 is the signature of L8a17. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                 1 1 -5               2 1 -1 -7             2     2 -9           2 2     0 -11         4 2       2 -13       1 3         2 -15     3 3           0 -17   1 3             2 -19   1               -1 -21 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{3}$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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, 17]]
Out[2]=	8
In[3]:=	PD[Link[8, Alternating, 17]]
Out[3]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[10, 13, 5, 14], X[8, 15, 9, 16], X[14, 7, 15, 8], X[16, 9, 11, 10], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[4]:=	GaussCode[Link[8, Alternating, 17]]
Out[4]=	GaussCode[{1, -7, 2, -8}, {7, -1, 5, -4, 6, -3}, {8, -2, 3, -5, 4, -6}]
In[5]:=	BR[Link[8, Alternating, 17]]
Out[5]=	BR[Link[8, Alternating, 17]]
In[6]:=	alex = Alexander[Link[8, Alternating, 17]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, Alternating, 17]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, Alternating, 17]], KnotSignature[Link[8, Alternating, 17]]}
Out[9]=	{Infinity, -4}
In[10]:=	J=Jones[Link[8, Alternating, 17]][q]
Out[10]=	-10 2 4 4 6 4 4 2 -2 q - -- + -- - -- + -- - -- + -- - -- + q 9 8 7 6 5 4 3 q 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, Alternating, 17]][q]
Out[12]=	-32 2 -28 3 3 3 5 3 4 -14 -10 q + --- + q + --- + --- + --- + --- + --- + --- + q + q - 30 26 24 22 20 18 16 q q q q q q q -8 -6 q + q
In[13]:=	Kauffman[Link[8, Alternating, 17]][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 4 2 6 2 8 2 10 2 12 2 6 a z - 2 a z + 10 a z + 15 a z + a z - 2 a z - 5 3 7 3 9 3 11 3 4 4 6 4 8 4 3 a z - 6 a z - 6 a z - 3 a z + a z - 8 a z - 12 a z - 10 4 12 4 5 5 7 5 9 5 11 5 2 a z + a z + 2 a z + 2 a z + 2 a z + 2 a z + 6 6 8 6 10 6 7 7 9 7 3 a z + 5 a z + 2 a z + a z + a z
In[14]:=	{Vassiliev[2][Link[8, Alternating, 17]], Vassiliev[3][Link[8, Alternating, 17]]}
Out[14]=	100 {0, ---} 3
In[15]:=	Kh[Link[8, Alternating, 17]][q, t]
Out[15]=	-5 -3 1 1 1 3 3 3 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 21 8 19 7 17 7 17 6 15 6 15 5 q t q t q t q t q t q t 1 3 4 2 2 2 2 2 ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- 13 5 13 4 11 4 11 3 9 3 9 2 7 2 5 q t q t q t q t q t q t q t q t