L9a25

L9a24

L9a26

	Visit L9a25's page at Knotilus! Visit L9a25's page at the original Knot Atlas!
	L9a25 is $9_{8}^{2}$ in the Rolfsen table of links.

L9a25 Further Notes and Views

Knot presentations

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

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, ${\frac {79}{48}}$)

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:L9a25/V 2,1 Data:L9a25/V 3,1 Data:L9a25/V 4,1 Data:L9a25/V 4,2 Data:L9a25/V 4,3 Data:L9a25/V 5,1 Data:L9a25/V 5,2 Data:L9a25/V 5,3 Data:L9a25/V 5,4 Data:L9a25/V 6,1 Data:L9a25/V 6,2 Data:L9a25/V 6,3 Data:L9a25/V 6,4 Data:L9a25/V 6,5 Data:L9a25/V 6,6 Data:L9a25/V 6,7 Data:L9a25/V 6,8 Data:L9a25/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 L9a25. 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 4                   1 1 2                 1   -1 0               3 1   2 -2             3 2     -1 -4           3 2       1 -6         3 4         1 -8       2 2           0 -10     1 3             2 -12   1 2               -1 -14   1                 1 -16 1                   -1

Integral Khovanov Homology (db, data source)

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

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, 25]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 25]]
Out[3]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[14, 6, 15, 5], X[18, 11, 7, 12], X[16, 13, 17, 14], X[12, 17, 13, 18], X[4, 16, 5, 15], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[4]:=	GaussCode[Link[9, Alternating, 25]]
Out[4]=	GaussCode[{1, -8, 2, -7, 3, -9}, {8, -1, 9, -2, 4, -6, 5, -3, 7, -5, 6, -4}]
In[5]:=	BR[Link[9, Alternating, 25]]
Out[5]=	BR[Link[9, Alternating, 25]]
In[6]:=	alex = Alexander[Link[9, Alternating, 25]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 25]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 25]], KnotSignature[Link[9, Alternating, 25]]}
Out[9]=	{Infinity, -1}
In[10]:=	J=Jones[Link[9, Alternating, 25]][q]
Out[10]=	-(15/2) 2 3 5 5 6 5 4 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 2 Sqrt[q] - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 25]][q]
Out[12]=	-24 -20 -18 -16 -14 2 -8 2 -4 2 6 2 - q + q - q + q + q + --- + q + -- + q - q + q 10 6 q q
In[13]:=	Kauffman[Link[9, Alternating, 25]][a, z]
Out[13]=	3 2 a a z 3 5 7 2 2 2 -a + - + -- + - - 3 a z - 5 a z + 2 a z + 3 a z + z - a z - z z a 3 4 2 6 2 8 2 z 3 3 3 5 3 a z - 3 a z - 4 a z - -- + 3 a z + 6 a z - 7 a z - a 7 3 4 2 4 4 4 6 4 8 4 5 9 a z - 2 z + 3 a z + 5 a z + 4 a z + 4 a z - 3 a z + 5 5 7 5 2 6 4 6 6 6 8 6 3 7 11 a z + 8 a z - 3 a z - a z + a z - a z - 2 a z - 5 7 7 7 4 8 6 8 4 a z - 2 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 25]], Vassiliev[3][Link[9, Alternating, 25]]}
Out[14]=	79 {0, --} 48
In[15]:=	Kh[Link[9, Alternating, 25]][q, t]
Out[15]=	2 1 1 1 2 1 3 2 3 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 q q t q t q t q t q t q t q t 2 3 4 3 2 3 2 4 2 ----- + ----- + ----- + ----- + ---- + ---- + t + q t + q t 8 3 6 3 6 2 4 2 4 2 q t q t q t q t q t q t