L9a37

L9a36

L9a38

	Visit L9a37's page at Knotilus! Visit L9a37's page at the original Knot Atlas!
	L9a37 is $9_{7}^{2}$ in the Rolfsen table of links.

L9a37 Further Notes and Views

Knot presentations

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X18,12,9,11 X2,9,3,10 X4,18,5,17 X16,8,17,7 X14,6,15,5 X6,16,7,15 X8,14,1,13
Gauss code	{1, -4, 2, -5, 7, -8, 6, -9}, {4, -1, 3, -2, 9, -7, 8, -6, 5, -3}

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, ${\frac {41}{24}}$)

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:L9a37/V 2,1 Data:L9a37/V 3,1 Data:L9a37/V 4,1 Data:L9a37/V 4,2 Data:L9a37/V 4,3 Data:L9a37/V 5,1 Data:L9a37/V 5,2 Data:L9a37/V 5,3 Data:L9a37/V 5,4 Data:L9a37/V 6,1 Data:L9a37/V 6,2 Data:L9a37/V 6,3 Data:L9a37/V 6,4 Data:L9a37/V 6,5 Data:L9a37/V 6,6 Data:L9a37/V 6,7 Data:L9a37/V 6,8 Data:L9a37/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 L9a37. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 χ 18                   1 -1 16                 2   2 14               3 1   -2 12             4 2     2 10           4 4       0 8         4 3         1 6       2 4           2 4     2 4             -2 2   1 3               2 0   1                 -1 -2 1                   1

Integral Khovanov Homology (db, data source)

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

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, 37]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 37]]
Out[3]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[18, 12, 9, 11], X[2, 9, 3, 10], X[4, 18, 5, 17], X[16, 8, 17, 7], X[14, 6, 15, 5], X[6, 16, 7, 15], X[8, 14, 1, 13]]
In[4]:=	GaussCode[Link[9, Alternating, 37]]
Out[4]=	GaussCode[{1, -4, 2, -5, 7, -8, 6, -9}, {4, -1, 3, -2, 9, -7, 8, -6, 5, -3}]
In[5]:=	BR[Link[9, Alternating, 37]]
Out[5]=	BR[Link[9, Alternating, 37]]
In[6]:=	alex = Alexander[Link[9, Alternating, 37]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 37]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 37]], KnotSignature[Link[9, Alternating, 37]]}
Out[9]=	{Infinity, 3}
In[10]:=	J=Jones[Link[9, Alternating, 37]][q]
Out[10]=	1 3/2 5/2 7/2 9/2 11/2 -(-------) + 2 Sqrt[q] - 4 q + 6 q - 8 q + 7 q - 7 q + Sqrt[q] 13/2 15/2 17/2 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, 37]][q]
Out[12]=	-2 4 6 8 10 12 14 18 20 24 26 q + q - 2 q + 2 q + q + 2 q + 3 q + 2 q - q + q - q
In[13]:=	Kauffman[Link[9, Alternating, 37]][a, z]
Out[13]=	2 2 2 2 -4 1 1 z 5 z 2 z 2 z z z 3 z 4 z a - ---- - ---- - -- + --- + --- - --- + --- - -- - ---- - ---- - 5 3 9 5 3 a 10 8 6 4 a z a z a a a a a a a 2 3 3 3 3 4 4 4 4 4 3 z 4 z 8 z z 3 z z 5 z 7 z 6 z 5 z ---- + ---- - ---- - -- + ---- - --- + ---- + ---- + ---- + ---- - 2 9 5 3 a 10 8 6 4 2 a a a a a a a a a 5 5 5 5 5 6 6 6 6 7 3 z 2 z 9 z 3 z z 4 z 4 z 2 z 2 z 3 z ---- + ---- + ---- + ---- - -- - ---- - ---- - ---- - ---- - ---- - 9 7 5 3 a 8 6 4 2 7 a a a a a a a a a 7 7 8 8 5 z 2 z z z ---- - ---- - -- - -- 5 3 6 4 a a a a
In[14]:=	{Vassiliev[2][Link[9, Alternating, 37]], Vassiliev[3][Link[9, Alternating, 37]]}
Out[14]=	41 {0, --} 24
In[15]:=	Kh[Link[9, Alternating, 37]][q, t]
Out[15]=	2 2 4 1 1 q 4 6 6 2 8 2 3 q + 2 q + ----- + - + -- + 4 q t + 2 q t + 4 q t + 4 q t + 2 2 t t q t 8 3 10 3 10 4 12 4 12 5 14 5 3 q t + 4 q t + 4 q t + 4 q t + 2 q t + 3 q t + 14 6 16 6 18 7 q t + 2 q t + q t