L9a46

L9a45

L9a47

	Visit L9a46's page at Knotilus! Visit L9a46's page at the original Knot Atlas!
	L9a46 is $9_{10}^{3}$ in the Rolfsen table of links.

L9a46 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X16,12,17,11 X18,9,11,10 X10,17,5,18 X4,15,1,16
Gauss code	{1, -4, 3, -9}, {2, -1, 5, -3, 7, -8}, {6, -2, 4, -5, 9, -6, 8, -7}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)^{2}(w-1)^{2}}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{-5}+q^{4}+4q^{-4}-4q^{3}-6q^{-3}+7q^{2}+10q^{-2}-9q-10q^{-1}+12$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{4}z^{2}+a^{4}z^{-2}+2a^{2}z^{4}+z^{4}a^{-2}+3a^{2}z^{2}+z^{2}a^{-2}-2a^{2}z^{-2}-a^{2}-z^{6}-3z^{4}-3z^{2}+z^{-2}+1$ (db)
Kauffman polynomial	$a^{5}z^{5}-a^{5}z^{3}+4a^{4}z^{6}-8a^{4}z^{4}+z^{4}a^{-4}+4a^{4}z^{2}-a^{4}z^{-2}+a^{4}+5a^{3}z^{7}-8a^{3}z^{5}+4z^{5}a^{-3}+2a^{3}z^{3}-3z^{3}a^{-3}-a^{3}z+2a^{3}z^{-1}+2a^{2}z^{8}+8a^{2}z^{6}+7z^{6}a^{-2}-23a^{2}z^{4}-9z^{4}a^{-2}+12a^{2}z^{2}+4z^{2}a^{-2}-2a^{2}z^{-2}+a^{2}+11az^{7}+6z^{7}a^{-1}-17az^{5}-4z^{5}a^{-1}+4az^{3}-2z^{3}a^{-1}-az+2az^{-1}+2z^{8}+11z^{6}-25z^{4}+12z^{2}-z^{-2}+1$ (db)

Vassiliev invariants

V2 and V3:

(0, ${\frac {5}{6}}$)

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:L9a46/V 2,1 Data:L9a46/V 3,1 Data:L9a46/V 4,1 Data:L9a46/V 4,2 Data:L9a46/V 4,3 Data:L9a46/V 5,1 Data:L9a46/V 5,2 Data:L9a46/V 5,3 Data:L9a46/V 5,4 Data:L9a46/V 6,1 Data:L9a46/V 6,2 Data:L9a46/V 6,3 Data:L9a46/V 6,4 Data:L9a46/V 6,5 Data:L9a46/V 6,6 Data:L9a46/V 6,7 Data:L9a46/V 6,8 Data:L9a46/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=$0 is the signature of L9a46. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                   1 1 7                 3   -3 5               4 1   3 3             5 3     -2 1           7 4       3 -1         5 7         2 -3       5 5           0 -5     3 7             4 -7   1 3               -2 -9   3                 3 -11 1                   -1

Integral Khovanov Homology (db, data source)

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

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, 46]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 46]]
Out[3]=	PD[X[6, 1, 7, 2], X[12, 6, 13, 5], X[8, 4, 9, 3], X[2, 14, 3, 13], X[14, 7, 15, 8], X[16, 12, 17, 11], X[18, 9, 11, 10], X[10, 17, 5, 18], X[4, 15, 1, 16]]
In[4]:=	GaussCode[Link[9, Alternating, 46]]
Out[4]=	GaussCode[{1, -4, 3, -9}, {2, -1, 5, -3, 7, -8}, {6, -2, 4, -5, 9, -6, 8, -7}]
In[5]:=	BR[Link[9, Alternating, 46]]
Out[5]=	BR[Link[9, Alternating, 46]]
In[6]:=	alex = Alexander[Link[9, Alternating, 46]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 46]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 46]], KnotSignature[Link[9, Alternating, 46]]}
Out[9]=	{Infinity, 0}
In[10]:=	J=Jones[Link[9, Alternating, 46]][q]
Out[10]=	-5 4 6 10 10 2 3 4 12 - q + -- - -- + -- - -- - 9 q + 7 q - 4 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[9, Alternating, 46]][q]
Out[12]=	-16 -14 3 -10 6 4 4 6 2 4 8 1 - q + q + --- + q + -- + -- + -- + -- + 4 q - 2 q + q - 12 8 6 4 2 q q q q q 10 12 2 q + q
In[13]:=	Kauffman[Link[9, Alternating, 46]][a, z]
Out[13]=	2 4 3 2 4 -2 2 a a 2 a 2 a 3 2 1 + a + a - z - ---- - -- + --- + ---- - a z - a z + 12 z + 2 2 z z z z 2 3 3 4 z 2 2 4 2 3 z 2 z 3 3 3 5 3 ---- + 12 a z + 4 a z - ---- - ---- + 4 a z + 2 a z - a z - 2 3 a a a 4 4 5 5 4 z 9 z 2 4 4 4 4 z 4 z 5 25 z + -- - ---- - 23 a z - 8 a z + ---- - ---- - 17 a z - 4 2 3 a a a a 6 7 3 5 5 5 6 7 z 2 6 4 6 6 z 7 8 a z + a z + 11 z + ---- + 8 a z + 4 a z + ---- + 11 a z + 2 a a 3 7 8 2 8 5 a z + 2 z + 2 a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 46]], Vassiliev[3][Link[9, Alternating, 46]]}
Out[14]=	5 {0, -} 6
In[15]:=	Kh[Link[9, Alternating, 46]][q, t]
Out[15]=	7 1 3 1 3 3 7 5 - + 7 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 5 5 3 3 2 5 2 5 3 7 3 ---- + --- + 4 q t + 5 q t + 3 q t + 4 q t + q t + 3 q t + 3 q t q t 9 4 q t