L9a53

L9a52

L9a54

	Visit L9a53's page at Knotilus! Visit L9a53's page at the original Knot Atlas!
	L9a53 is $9_{12}^{3}$ in the Rolfsen table of links.

L9a53 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X18,10,15,9 X8493 X16,5,17,6 X14,17,5,18 X10,16,11,15 X2,12,3,11
Gauss code	{1, -9, 5, -3}, {8, -6, 7, -4}, {6, -1, 2, -5, 4, -8, 9, -2, 3, -7}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)^{3}}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{4}-4q^{3}+8q^{2}-9q+12-10q^{-1}+10q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{4}z^{2}-a^{4}+2a^{2}z^{4}+z^{4}a^{-2}+4a^{2}z^{2}+z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+3a^{2}+a^{-2}-z^{6}-3z^{4}-4z^{2}-2z^{-2}-3$ (db)
Kauffman polynomial	$2a^{2}z^{8}+2z^{8}+4a^{3}z^{7}+11az^{7}+7z^{7}a^{-1}+3a^{4}z^{6}+7a^{2}z^{6}+8z^{6}a^{-2}+12z^{6}+a^{5}z^{5}-5a^{3}z^{5}-17az^{5}-7z^{5}a^{-1}+4z^{5}a^{-3}-6a^{4}z^{4}-22a^{2}z^{4}-11z^{4}a^{-2}+z^{4}a^{-4}-28z^{4}-2a^{5}z^{3}-a^{3}z^{3}+3az^{3}-2z^{3}a^{-3}+5a^{4}z^{2}+17a^{2}z^{2}+5z^{2}a^{-2}+17z^{2}+a^{5}z+3a^{3}z+3az+za^{-1}-2a^{4}-6a^{2}-2a^{-2}-5-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (db)

Vassiliev invariants

V2 and V3:

(0, -1)

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:L9a53/V 2,1 Data:L9a53/V 3,1 Data:L9a53/V 4,1 Data:L9a53/V 4,2 Data:L9a53/V 4,3 Data:L9a53/V 5,1 Data:L9a53/V 5,2 Data:L9a53/V 5,3 Data:L9a53/V 5,4 Data:L9a53/V 6,1 Data:L9a53/V 6,2 Data:L9a53/V 6,3 Data:L9a53/V 6,4 Data:L9a53/V 6,5 Data:L9a53/V 6,6 Data:L9a53/V 6,7 Data:L9a53/V 6,8 Data:L9a53/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=$0 is the signature of L9a53. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                   1 1 7                 3   -3 5               5 1   4 3             4 3     -1 1           8 5       3 -1         6 8         2 -3       4 4           0 -5     2 6             4 -7   1 4               -3 -9   2                 2 -11 1                   -1

Integral Khovanov Homology (db, data source)

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

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[2, Alternating, 1]]
Out[2]=	2
In[3]:=	PD[Link[2, Alternating, 1]]
Out[3]=	PD[X[4, 1, 3, 2], X[2, 3, 1, 4]]
In[4]:=	GaussCode[Link[2, Alternating, 1]]
Out[4]=	GaussCode[{1, -2}, {2, -1}]
In[5]:=	BR[Link[2, Alternating, 1]]
Out[5]=	BR[Link[2, Alternating, 1]]
In[6]:=	alex = Alexander[Link[2, Alternating, 1]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[2, Alternating, 1]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[2, Alternating, 1]], KnotSignature[Link[2, Alternating, 1]]}
Out[9]=	{Infinity, -1}
In[10]:=	J=Jones[Link[2, Alternating, 1]][q]
Out[10]=	-(5/2) 1 -q - ------- Sqrt[q]
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[2, Alternating, 1]][q]
Out[12]=	-10 2 2 2 -2 1 + q + -- + -- + -- + q 8 6 4 q q q
In[13]:=	Kauffman[Link[2, Alternating, 1]][a, z]
Out[13]=	3 2 a a 3 -a + - + -- - a z - a z z z
In[14]:=	{Vassiliev[2][Link[2, Alternating, 1]], Vassiliev[3][Link[2, Alternating, 1]]}
Out[14]=	17 {0, -(--)} 48
In[15]:=	Kh[Link[2, Alternating, 1]][q, t]
Out[15]=	-2 1 1 1 + q + ----- + ----- 6 2 4 2 q t q t