L9a27

L9a26

L9a28

	Visit L9a27's page at Knotilus! Visit L9a27's page at the original Knot Atlas!
	L9a27 is $9_{12}^{2}$ in the Rolfsen table of links.

L9a27 Further Notes and Views

Knot presentations

Planar diagram presentation	X8192 X10,4,11,3 X18,10,7,9 X2738 X16,13,17,14 X6,12,1,11 X4,16,5,15 X14,6,15,5 X12,17,13,18
Gauss code	{1, -4, 2, -7, 8, -6}, {4, -1, 3, -2, 6, -9, 5, -8, 7, -5, 9, -3}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(uv-u-2v+1)(uv-2u-v+1)}{uv}}$ (db)
Jones polynomial	$q^{11/2}-3q^{9/2}+5q^{7/2}-8q^{5/2}+8q^{3/2}-9{\sqrt {q}}+{\frac {7}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-5}-2z^{3}a^{-3}+a^{3}z-2za^{-3}-a^{-3}z^{-1}+z^{5}a^{-1}-2az^{3}+2z^{3}a^{-1}-2az+3za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-2}-z^{8}-3az^{7}-6z^{7}a^{-1}-3z^{7}a^{-3}-3a^{2}z^{6}-7z^{6}a^{-2}-4z^{6}a^{-4}-6z^{6}-a^{3}z^{5}+4az^{5}+7z^{5}a^{-1}-z^{5}a^{-3}-3z^{5}a^{-5}+7a^{2}z^{4}+13z^{4}a^{-2}+4z^{4}a^{-4}-z^{4}a^{-6}+15z^{4}+2a^{3}z^{3}+az^{3}+z^{3}a^{-1}+6z^{3}a^{-3}+4z^{3}a^{-5}-4a^{2}z^{2}-5z^{2}a^{-2}-z^{2}a^{-4}+z^{2}a^{-6}-7z^{2}-a^{3}z-az-2za^{-1}-4za^{-3}-2za^{-5}-a^{-2}+a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)

Vassiliev invariants

V2 and V3:

(0, $-{\frac {31}{48}}$)

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:L9a27/V 2,1 Data:L9a27/V 3,1 Data:L9a27/V 4,1 Data:L9a27/V 4,2 Data:L9a27/V 4,3 Data:L9a27/V 5,1 Data:L9a27/V 5,2 Data:L9a27/V 5,3 Data:L9a27/V 5,4 Data:L9a27/V 6,1 Data:L9a27/V 6,2 Data:L9a27/V 6,3 Data:L9a27/V 6,4 Data:L9a27/V 6,5 Data:L9a27/V 6,6 Data:L9a27/V 6,7 Data:L9a27/V 6,8 Data:L9a27/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 L9a27. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 12                   1 -1 10                 2   2 8               3 1   -2 6             5 2     3 4           4 4       0 2         5 4         1 0       3 5           2 -2     2 4             -2 -4   1 3               2 -6   2                 -2 -8 1                   1

Integral Khovanov Homology (db, data source)

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

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, 27]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 27]]
Out[3]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[18, 10, 7, 9], X[2, 7, 3, 8], X[16, 13, 17, 14], X[6, 12, 1, 11], X[4, 16, 5, 15], X[14, 6, 15, 5], X[12, 17, 13, 18]]
In[4]:=	GaussCode[Link[9, Alternating, 27]]
Out[4]=	GaussCode[{1, -4, 2, -7, 8, -6}, {4, -1, 3, -2, 6, -9, 5, -8, 7, -5, 9, -3}]
In[5]:=	BR[Link[9, Alternating, 27]]
Out[5]=	BR[Link[9, Alternating, 27]]
In[6]:=	alex = Alexander[Link[9, Alternating, 27]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 27]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 27]], KnotSignature[Link[9, Alternating, 27]]}
Out[9]=	{Infinity, 1}
In[10]:=	J=Jones[Link[9, Alternating, 27]][q]
Out[10]=	-(7/2) 3 5 7 3/2 5/2 -q + ---- - ---- + ------- - 9 Sqrt[q] + 8 q - 8 q + 5/2 3/2 Sqrt[q] q q 7/2 9/2 11/2 5 q - 3 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 27]][q]
Out[12]=	-12 2 -6 -4 4 8 10 12 14 18 3 + q - -- + q - q + 3 q + 2 q + 2 q - q + 2 q - q 8 q
In[13]:=	Kauffman[Link[9, Alternating, 27]][a, z]
Out[13]=	2 2 -2 1 1 2 z 4 z 2 z 3 2 z z -a + ---- + --- - --- - --- - --- - a z - a z - 7 z + -- - -- - 3 a z 5 3 a 6 4 a z a a a a 2 3 3 3 4 5 z 2 2 4 z 6 z z 3 3 3 4 z ---- - 4 a z + ---- + ---- + -- + a z + 2 a z + 15 z - -- + 2 5 3 a 6 a a a a 4 4 5 5 5 4 z 13 z 2 4 3 z z 7 z 5 3 5 6 ---- + ----- + 7 a z - ---- - -- + ---- + 4 a z - a z - 6 z - 4 2 5 3 a a a a a 6 6 7 7 8 4 z 7 z 2 6 3 z 6 z 7 8 z ---- - ---- - 3 a z - ---- - ---- - 3 a z - z - -- 4 2 3 a 2 a a a a
In[14]:=	{Vassiliev[2][Link[9, Alternating, 27]], Vassiliev[3][Link[9, Alternating, 27]]}
Out[14]=	31 {0, -(--)} 48
In[15]:=	Kh[Link[9, Alternating, 27]][q, t]
Out[15]=	2 1 2 1 3 2 3 4 2 5 + 5 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 4 q t + 8 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 4 2 6 2 6 3 8 3 8 4 10 4 4 q t + 4 q t + 5 q t + 2 q t + 3 q t + q t + 2 q t + 12 5 q t