L9a51

L9a50

L9a52

	Visit L9a51's page at Knotilus! Visit L9a51's page at the original Knot Atlas!
	L9a51 is $9_{11}^{3}$ in the Rolfsen table of links.

L9a51 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X12,4,13,3 X8,12,9,11 X18,8,11,7 X16,13,17,14 X14,6,15,5 X10,16,5,15 X2,9,3,10 X4,18,1,17
Gauss code	{1, -8, 2, -9}, {6, -1, 4, -3, 8, -7}, {3, -2, 5, -6, 7, -5, 9, -4}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{2}w^{2}-2uv^{2}w-2uvw^{2}+4uvw-2uv+uw^{2}-2uw+u-v^{2}w^{2}+2v^{2}w-v^{2}+2vw^{2}-4vw+2v+2w-1}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$q^{7}-3q^{6}+7q^{5}-8q^{4}+11q^{3}-10q^{2}-q^{-2}+9q+4q^{-1}-6$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{2}a^{-6}+a^{-6}z^{-2}+a^{-6}-2z^{4}a^{-4}-4z^{2}a^{-4}-2a^{-4}z^{-2}-3a^{-4}+z^{6}a^{-2}+3z^{4}a^{-2}+3z^{2}a^{-2}+a^{-2}z^{-2}+a^{-2}-z^{4}-z^{2}+1$ (db)
Kauffman polynomial	$z^{4}a^{-8}-z^{2}a^{-8}+3z^{5}a^{-7}-2z^{3}a^{-7}+6z^{6}a^{-6}-10z^{4}a^{-6}+10z^{2}a^{-6}+a^{-6}z^{-2}-5a^{-6}+5z^{7}a^{-5}-3z^{5}a^{-5}-4z^{3}a^{-5}+6za^{-5}-2a^{-5}z^{-1}+2z^{8}a^{-4}+8z^{6}a^{-4}-21z^{4}a^{-4}+17z^{2}a^{-4}+2a^{-4}z^{-2}-8a^{-4}+10z^{7}a^{-3}-16z^{5}a^{-3}+2z^{3}a^{-3}+6za^{-3}-2a^{-3}z^{-1}+2z^{8}a^{-2}+6z^{6}a^{-2}-18z^{4}a^{-2}+9z^{2}a^{-2}+a^{-2}z^{-2}-3a^{-2}+5z^{7}a^{-1}+az^{5}-9z^{5}a^{-1}-az^{3}+3z^{3}a^{-1}+4z^{6}-8z^{4}+3z^{2}+1$ (db)

Vassiliev invariants

V2 and V3:

(0, $-{\frac {23}{3}}$)

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:L9a51/V 2,1 Data:L9a51/V 3,1 Data:L9a51/V 4,1 Data:L9a51/V 4,2 Data:L9a51/V 4,3 Data:L9a51/V 5,1 Data:L9a51/V 5,2 Data:L9a51/V 5,3 Data:L9a51/V 5,4 Data:L9a51/V 6,1 Data:L9a51/V 6,2 Data:L9a51/V 6,3 Data:L9a51/V 6,4 Data:L9a51/V 6,5 Data:L9a51/V 6,6 Data:L9a51/V 6,7 Data:L9a51/V 6,8 Data:L9a51/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 L9a51. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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[9, Alternating, 51]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 51]]
Out[3]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[8, 12, 9, 11], X[18, 8, 11, 7], X[16, 13, 17, 14], X[14, 6, 15, 5], X[10, 16, 5, 15], X[2, 9, 3, 10], X[4, 18, 1, 17]]
In[4]:=	GaussCode[Link[9, Alternating, 51]]
Out[4]=	GaussCode[{1, -8, 2, -9}, {6, -1, 4, -3, 8, -7}, {3, -2, 5, -6, 7, -5, 9, -4}]
In[5]:=	BR[Link[9, Alternating, 51]]
Out[5]=	BR[Link[9, Alternating, 51]]
In[6]:=	alex = Alexander[Link[9, Alternating, 51]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 51]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 51]], KnotSignature[Link[9, Alternating, 51]]}
Out[9]=	{Infinity, 2}
In[10]:=	J=Jones[Link[9, Alternating, 51]][q]
Out[10]=	-2 4 2 3 4 5 6 7 -6 - q + - + 9 q - 10 q + 11 q - 8 q + 7 q - 3 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 51]][q]
Out[12]=	-6 2 2 4 6 8 10 12 14 1 - q + -- + 3 q - 2 q + 3 q + q + 5 q + 6 q + 3 q + 4 q 16 22 5 q + q
In[13]:=	Kauffman[Link[9, Alternating, 51]][a, z]
Out[13]=	5 8 3 1 2 1 2 2 6 z 6 z 1 - -- - -- - -- + ----- + ----- + ----- - ---- - ---- + --- + --- + 6 4 2 6 2 4 2 2 2 5 3 5 3 a a a a z a z a z a z a z a a 2 2 2 2 3 3 3 3 2 z 10 z 17 z 9 z 2 z 4 z 2 z 3 z 3 3 z - -- + ----- + ----- + ---- - ---- - ---- + ---- + ---- - a z - 8 6 4 2 7 5 3 a a a a a a a a 4 4 4 4 5 5 5 5 4 z 10 z 21 z 18 z 3 z 3 z 16 z 9 z 8 z + -- - ----- - ----- - ----- + ---- - ---- - ----- - ---- + 8 6 4 2 7 5 3 a a a a a a a a 6 6 6 7 7 7 8 8 5 6 6 z 8 z 6 z 5 z 10 z 5 z 2 z 2 z a z + 4 z + ---- + ---- + ---- + ---- + ----- + ---- + ---- + ---- 6 4 2 5 3 a 4 2 a a a a a a a
In[14]:=	{Vassiliev[2][Link[9, Alternating, 51]], Vassiliev[3][Link[9, Alternating, 51]]}
Out[14]=	23 {0, -(--)} 3
In[15]:=	Kh[Link[9, Alternating, 51]][q, t]
Out[15]=	3 1 3 1 3 3 q 3 5 6 q + 5 q + ----- + ----- + ---- + --- + --- + 6 q t + 4 q t + 5 3 3 2 2 q t t q t q t q t 5 2 7 2 7 3 9 3 9 4 11 4 5 q t + 7 q t + 4 q t + 4 q t + 3 q t + 4 q t + 13 5 13 6 15 6 3 q t + q t + q t