L9a16

L9a15

L9a17

	Visit L9a16's page at Knotilus! Visit L9a16's page at the original Knot Atlas!
	L9a16 is $9_{28}^{2}$ in the Rolfsen table of links.

L9a16 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{3}-4uv^{2}+4uv-2u-2v^{3}+4v^{2}-4v+1}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {6}{q^{5/2}}}-{\frac {8}{q^{7/2}}}+{\frac {7}{q^{9/2}}}-{\frac {7}{q^{11/2}}}+{\frac {4}{q^{13/2}}}-{\frac {2}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{9}z^{-1}+3za^{7}+3a^{7}z^{-1}-3z^{3}a^{5}-5za^{5}-2a^{5}z^{-1}+z^{5}a^{3}+2z^{3}a^{3}+za^{3}-z^{3}a-za$ (db)
Kauffman polynomial	$a^{10}z^{4}-2a^{10}z^{2}+a^{10}+2a^{9}z^{5}-3a^{9}z^{3}+2a^{9}z-a^{9}z^{-1}+2a^{8}z^{6}+a^{8}z^{4}-5a^{8}z^{2}+3a^{8}+2a^{7}z^{7}+a^{7}z^{5}-5a^{7}z^{3}+7a^{7}z-3a^{7}z^{-1}+a^{6}z^{8}+3a^{6}z^{6}-4a^{6}z^{4}-a^{6}z^{2}+3a^{6}+5a^{5}z^{7}-7a^{5}z^{5}+4a^{5}z-2a^{5}z^{-1}+a^{4}z^{8}+4a^{4}z^{6}-11a^{4}z^{4}+5a^{4}z^{2}+3a^{3}z^{7}-5a^{3}z^{5}+3a^{2}z^{6}-7a^{2}z^{4}+3a^{2}z^{2}+az^{5}-2az^{3}+az$ (db)

Vassiliev invariants

V2 and V3:

(0, ${\frac {55}{24}}$)

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:L9a16/V 2,1 Data:L9a16/V 3,1 Data:L9a16/V 4,1 Data:L9a16/V 4,2 Data:L9a16/V 4,3 Data:L9a16/V 5,1 Data:L9a16/V 5,2 Data:L9a16/V 5,3 Data:L9a16/V 5,4 Data:L9a16/V 6,1 Data:L9a16/V 6,2 Data:L9a16/V 6,3 Data:L9a16/V 6,4 Data:L9a16/V 6,5 Data:L9a16/V 6,6 Data:L9a16/V 6,7 Data:L9a16/V 6,8 Data:L9a16/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 $s=$-3 is the signature of L9a16. 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 2                   1 1 0                 2   -2 -2               3 1   2 -4             4 3     -1 -6           4 2       2 -8         3 4         1 -10       4 4           0 -12     1 4             3 -14   1 3               -2 -16   1                 1 -18 1                   -1

Integral Khovanov Homology (db, data source)

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

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, 16]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 16]]
Out[3]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[14, 10, 15, 9], X[10, 14, 11, 13], X[18, 15, 5, 16], X[16, 7, 17, 8], X[8, 17, 9, 18], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[4]:=	GaussCode[Link[9, Alternating, 16]]
Out[4]=	GaussCode[{1, -8, 2, -9}, {8, -1, 6, -7, 3, -4, 9, -2, 4, -3, 5, -6, 7, -5}]
In[5]:=	BR[Link[9, Alternating, 16]]
Out[5]=	BR[Link[9, Alternating, 16]]
In[6]:=	alex = Alexander[Link[9, Alternating, 16]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 16]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 16]], KnotSignature[Link[9, Alternating, 16]]}
Out[9]=	{Infinity, -3}
In[10]:=	J=Jones[Link[9, Alternating, 16]][q]
Out[10]=	-(17/2) 2 4 7 7 8 6 5 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q 3 ------- - Sqrt[q] Sqrt[q]
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 16]][q]
Out[12]=	-28 2 4 2 3 2 2 2 -4 2 -1 - q - --- + --- + --- + --- + --- + -- - -- + q + q 26 18 16 14 12 8 6 q q q q q q q
In[13]:=	Kauffman[Link[9, Alternating, 16]][a, z]
Out[13]=	5 7 9 6 8 10 2 a 3 a a 5 7 -3 a - 3 a - a + ---- + ---- + -- - a z - 4 a z - 7 a z - z z z 9 2 2 4 2 6 2 8 2 10 2 3 2 a z - 3 a z - 5 a z + a z + 5 a z + 2 a z + 2 a z + 7 3 9 3 2 4 4 4 6 4 8 4 10 4 5 a z + 3 a z + 7 a z + 11 a z + 4 a z - a z - a z - 5 3 5 5 5 7 5 9 5 2 6 4 6 a z + 5 a z + 7 a z - a z - 2 a z - 3 a z - 4 a z - 6 6 8 6 3 7 5 7 7 7 4 8 6 8 3 a z - 2 a z - 3 a z - 5 a z - 2 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 16]], Vassiliev[3][Link[9, Alternating, 16]]}
Out[14]=	55 {0, --} 24
In[15]:=	Kh[Link[9, Alternating, 16]][q, t]
Out[15]=	3 3 1 1 1 3 1 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 q q q t q t q t q t q t q t 4 4 3 4 4 2 4 t ------ + ------ + ----- + ----- + ----- + ---- + ---- + 2 t + -- + 10 4 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q t q 2 2 q t