L9a52

L9a51

L9a53

	Visit L9a52's page at Knotilus! Visit L9a52's page at the original Knot Atlas!
	L9a52 is $9_{8}^{3}$ in the Rolfsen table of links.

L9a52 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X10,3,11,4 X16,11,17,12 X14,8,15,7 X8,14,9,13 X18,15,13,16 X12,17,5,18 X2536 X4,9,1,10
Gauss code	{1, -8, 2, -9}, {5, -4, 6, -3, 7, -6}, {8, -1, 4, -5, 9, -2, 3, -7}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(v-1)(w-1)\left(2uw-u+w^{2}-2w\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{-8}+3q^{-7}-5q^{-6}+8q^{-5}-7q^{-4}+9q^{-3}-7q^{-2}+q+5q^{-1}-2$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{8}+3a^{6}z^{2}+a^{6}z^{-2}+3a^{6}-2a^{4}z^{4}-4a^{4}z^{2}-2a^{4}z^{-2}-4a^{4}-a^{2}z^{4}+a^{2}z^{-2}+a^{2}+z^{2}+1$ (db)
Kauffman polynomial	$a^{9}z^{5}-2a^{9}z^{3}+a^{9}z+3a^{8}z^{6}-7a^{8}z^{4}+5a^{8}z^{2}-2a^{8}+3a^{7}z^{7}-3a^{7}z^{5}-5a^{7}z^{3}+3a^{7}z+a^{6}z^{8}+7a^{6}z^{6}-22a^{6}z^{4}+20a^{6}z^{2}+a^{6}z^{-2}-9a^{6}+6a^{5}z^{7}-8a^{5}z^{5}-2a^{5}z^{3}+7a^{5}z-2a^{5}z^{-1}+a^{4}z^{8}+7a^{4}z^{6}-19a^{4}z^{4}+20a^{4}z^{2}+2a^{4}z^{-2}-10a^{4}+3a^{3}z^{7}-2a^{3}z^{5}-a^{3}z^{3}+5a^{3}z-2a^{3}z^{-1}+3a^{2}z^{6}-3a^{2}z^{4}+3a^{2}z^{2}+a^{2}z^{-2}-3a^{2}+2az^{5}-2az^{3}+z^{4}-2z^{2}+1$ (db)

Vassiliev invariants

V2 and V3:

(0, ${\frac {14}{3}}$)

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:L9a52/V 2,1 Data:L9a52/V 3,1 Data:L9a52/V 4,1 Data:L9a52/V 4,2 Data:L9a52/V 4,3 Data:L9a52/V 5,1 Data:L9a52/V 5,2 Data:L9a52/V 5,3 Data:L9a52/V 5,4 Data:L9a52/V 6,1 Data:L9a52/V 6,2 Data:L9a52/V 6,3 Data:L9a52/V 6,4 Data:L9a52/V 6,5 Data:L9a52/V 6,6 Data:L9a52/V 6,7 Data:L9a52/V 6,8 Data:L9a52/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 L9a52. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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[9, Alternating, 52]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 52]]
Out[3]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 11, 17, 12], X[14, 8, 15, 7], X[8, 14, 9, 13], X[18, 15, 13, 16], X[12, 17, 5, 18], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[4]:=	GaussCode[Link[9, Alternating, 52]]
Out[4]=	GaussCode[{1, -8, 2, -9}, {5, -4, 6, -3, 7, -6}, {8, -1, 4, -5, 9, -2, 3, -7}]
In[5]:=	BR[Link[9, Alternating, 52]]
Out[5]=	BR[Link[9, Alternating, 52]]
In[6]:=	alex = Alexander[Link[9, Alternating, 52]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 52]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 52]], KnotSignature[Link[9, Alternating, 52]]}
Out[9]=	{Infinity, -2}
In[10]:=	J=Jones[Link[9, Alternating, 52]][q]
Out[10]=	-8 3 5 8 7 9 7 5 -2 - q + -- - -- + -- - -- + -- - -- + - + q 7 6 5 4 3 2 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[9, Alternating, 52]][q]
Out[12]=	-26 -24 2 2 5 3 6 3 3 2 -4 3 -q - q + --- + --- + --- + --- + --- + --- + -- + -- - q + -- + 22 18 16 14 12 10 8 6 2 q q q q q q q q q 4 q
In[13]:=	Kauffman[Link[9, Alternating, 52]][a, z]
Out[13]=	2 4 6 3 5 2 4 6 8 a 2 a a 2 a 2 a 1 - 3 a - 10 a - 9 a - 2 a + -- + ---- + -- - ---- - ---- + 2 2 2 z z z z z 3 5 7 9 2 2 2 4 2 5 a z + 7 a z + 3 a z + a z - 2 z + 3 a z + 20 a z + 6 2 8 2 3 3 3 5 3 7 3 9 3 20 a z + 5 a z - 2 a z - a z - 2 a z - 5 a z - 2 a z + 4 2 4 4 4 6 4 8 4 5 3 5 z - 3 a z - 19 a z - 22 a z - 7 a z + 2 a z - 2 a z - 5 5 7 5 9 5 2 6 4 6 6 6 8 6 8 a z - 3 a z + a z + 3 a z + 7 a z + 7 a z + 3 a z + 3 7 5 7 7 7 4 8 6 8 3 a z + 6 a z + 3 a z + a z + a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 52]], Vassiliev[3][Link[9, Alternating, 52]]}
Out[14]=	14 {0, --} 3
In[15]:=	Kh[Link[9, Alternating, 52]][q, t]
Out[15]=	3 4 1 2 1 3 2 5 5 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 q q t q t q t q t q t q t q t 4 3 5 4 2 5 t 3 2 ----- + ----- + ----- + ----- + ---- + ---- + - + q t + q t 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t