L9a31

L9a30

L9a32

	Visit L9a31's page at Knotilus! Visit L9a31's page at the original Knot Atlas!
	L9a31 is $9_{39}^{2}$ in the Rolfsen table of links.

L9a31 Further Notes and Views

Knot presentations

Planar diagram presentation	X8192 X16,9,17,10 X6718 X18,13,7,14 X10,4,11,3 X14,6,15,5 X4,12,5,11 X12,17,13,18 X2,16,3,15
Gauss code	{1, -9, 5, -7, 6, -3}, {3, -1, 2, -5, 7, -8, 4, -6, 9, -2, 8, -4}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {\left(v^{2}-v+1\right)(uv-u+1)(uv-v+1)}{uv^{2}}}$ (db)
Jones polynomial	$-q^{7/2}+3q^{5/2}-6q^{3/2}+7{\sqrt {q}}-{\frac {10}{\sqrt {q}}}+{\frac {9}{q^{3/2}}}-{\frac {8}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-3a^{3}z^{3}-3a^{3}z-a^{3}z^{-1}+az^{7}+5az^{5}-z^{5}a^{-1}+9az^{3}-3z^{3}a^{-1}+7az-3za^{-1}+3az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}-a^{6}z^{2}+3a^{5}z^{5}-3a^{5}z^{3}+5a^{4}z^{6}-7a^{4}z^{4}+4a^{4}z^{2}-a^{4}+5a^{3}z^{7}-8a^{3}z^{5}+z^{5}a^{-3}+8a^{3}z^{3}-2z^{3}a^{-3}-4a^{3}z+za^{-3}+a^{3}z^{-1}+2a^{2}z^{8}+4a^{2}z^{6}+3z^{6}a^{-2}-12a^{2}z^{4}-6z^{4}a^{-2}+10a^{2}z^{2}+2z^{2}a^{-2}-3a^{2}+9az^{7}+4z^{7}a^{-1}-20az^{5}-8z^{5}a^{-1}+18az^{3}+5z^{3}a^{-1}-9az-4za^{-1}+3az^{-1}+2a^{-1}z^{-1}+2z^{8}+2z^{6}-10z^{4}+7z^{2}-3$ (db)

Vassiliev invariants

V2 and V3:

(0, ${\frac {89}{48}}$)

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:L9a31/V 2,1 Data:L9a31/V 3,1 Data:L9a31/V 4,1 Data:L9a31/V 4,2 Data:L9a31/V 4,3 Data:L9a31/V 5,1 Data:L9a31/V 5,2 Data:L9a31/V 5,3 Data:L9a31/V 5,4 Data:L9a31/V 6,1 Data:L9a31/V 6,2 Data:L9a31/V 6,3 Data:L9a31/V 6,4 Data:L9a31/V 6,5 Data:L9a31/V 6,6 Data:L9a31/V 6,7 Data:L9a31/V 6,8 Data:L9a31/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 $s=$-1 is the signature of L9a31. 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 8                   1 1 6                 2   -2 4               4 1   3 2             4 3     -1 0           6 3       3 -2         4 5         1 -4       4 5           -1 -6     2 4             2 -8   1 4               -3 -10   2                 2 -12 1                   -1

Integral Khovanov Homology (db, data source)

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

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[9, Alternating, 31]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 31]]
Out[3]=	PD[X[8, 1, 9, 2], X[16, 9, 17, 10], X[6, 7, 1, 8], X[18, 13, 7, 14], X[10, 4, 11, 3], X[14, 6, 15, 5], X[4, 12, 5, 11], X[12, 17, 13, 18], X[2, 16, 3, 15]]
In[4]:=	GaussCode[Link[9, Alternating, 31]]
Out[4]=	GaussCode[{1, -9, 5, -7, 6, -3}, {3, -1, 2, -5, 7, -8, 4, -6, 9, -2, 8, -4}]
In[5]:=	BR[Link[9, Alternating, 31]]
Out[5]=	BR[Link[9, Alternating, 31]]
In[6]:=	alex = Alexander[Link[9, Alternating, 31]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 31]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 31]], KnotSignature[Link[9, Alternating, 31]]}
Out[9]=	{Infinity, -1}
In[10]:=	J=Jones[Link[9, Alternating, 31]][q]
Out[10]=	-(11/2) 3 6 8 9 10 3/2 q - ---- + ---- - ---- + ---- - ------- + 7 Sqrt[q] - 6 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 3 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 31]][q]
Out[12]=	-16 -14 2 2 3 2 4 6 8 10 5 - q + q - --- - -- + -- + 2 q + q + 2 q - q + q 12 6 4 q q q
In[13]:=	Kauffman[Link[9, Alternating, 31]][a, z]
Out[13]=	3 2 4 2 3 a a z 4 z 3 2 3 + 3 a + a - --- - --- - -- - -- + --- + 9 a z + 4 a z - 7 z - a z z z 3 a a 2 3 3 2 z 2 2 4 2 6 2 2 z 5 z 3 3 3 ---- - 10 a z - 4 a z + a z + ---- - ---- - 18 a z - 8 a z + 2 3 a a a 4 5 5 5 3 4 6 z 2 4 4 4 6 4 z 8 z 3 a z + 10 z + ---- + 12 a z + 7 a z - a z - -- + ---- + 2 3 a a a 6 5 3 5 5 5 6 3 z 2 6 4 6 20 a z + 8 a z - 3 a z - 2 z - ---- - 4 a z - 5 a z - 2 a 7 4 z 7 3 7 8 2 8 ---- - 9 a z - 5 a z - 2 z - 2 a z a
In[14]:=	{Vassiliev[2][Link[9, Alternating, 31]], Vassiliev[3][Link[9, Alternating, 31]]}
Out[14]=	89 {0, --} 48
In[15]:=	Kh[Link[9, Alternating, 31]][q, t]
Out[15]=	5 1 2 1 4 2 4 4 6 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 q q t q t q t q t q t q t q t 5 4 2 2 2 4 2 4 3 6 3 ---- + ---- + 3 t + 4 q t + 3 q t + 4 q t + q t + 2 q t + 4 2 q t q t 8 4 q t