L9a8

L9a7

L9a9

	Visit L9a8's page at Knotilus! Visit L9a8's page at the original Knot Atlas!
	L9a8 is $9_{25}^{2}$ in the Rolfsen table of links.

L9a8 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}-4t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{11/2}-3q^{9/2}+5q^{7/2}-8q^{5/2}+8q^{3/2}-8{\sqrt {q}}+{\frac {7}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {2}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-1}-2az^{3}+2z^{3}a^{-1}-2z^{3}a^{-3}+a^{3}z-3az+3za^{-1}-2za^{-3}+za^{-5}+a^{3}z^{-1}-2az^{-1}+2a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}-z^{2}a^{-6}+3z^{5}a^{-5}-4z^{3}a^{-5}+2za^{-5}+4z^{6}a^{-4}-4z^{4}a^{-4}+z^{2}a^{-4}+3z^{7}a^{-3}+a^{3}z^{5}+z^{5}a^{-3}-3a^{3}z^{3}-7z^{3}a^{-3}+3a^{3}z+5za^{-3}-a^{3}z^{-1}-a^{-3}z^{-1}+z^{8}a^{-2}+2a^{2}z^{6}+6z^{6}a^{-2}-4a^{2}z^{4}-10z^{4}a^{-2}+2a^{2}z^{2}+3z^{2}a^{-2}+2az^{7}+5z^{7}a^{-1}-3z^{5}a^{-1}-8az^{3}-8z^{3}a^{-1}+8az+8za^{-1}-2az^{-1}-2a^{-1}z^{-1}+z^{8}+4z^{6}-9z^{4}+3z^{2}-1$ (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:L9a8/V 2,1 Data:L9a8/V 3,1 Data:L9a8/V 4,1 Data:L9a8/V 4,2 Data:L9a8/V 4,3 Data:L9a8/V 5,1 Data:L9a8/V 5,2 Data:L9a8/V 5,3 Data:L9a8/V 5,4 Data:L9a8/V 6,1 Data:L9a8/V 6,2 Data:L9a8/V 6,3 Data:L9a8/V 6,4 Data:L9a8/V 6,5 Data:L9a8/V 6,6 Data:L9a8/V 6,7 Data:L9a8/V 6,8 Data:L9a8/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 L9a8. 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           3 3       0 2         5 5         0 0       4 5           1 -2     1 3             -2 -4   1 4               3 -6   1                 -1 -8 1                   1

Integral Khovanov Homology (db, data source)

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

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