L9a48

L9a47

L9a49

	Visit L9a48's page at Knotilus! Visit L9a48's page at the original Knot Atlas!
	L9a48 is $9_{5}^{3}$ in the Rolfsen table of links.

L9a48 Further Notes and Views

Knot presentations

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

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, 0)

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:L9a48/V 2,1 Data:L9a48/V 3,1 Data:L9a48/V 4,1 Data:L9a48/V 4,2 Data:L9a48/V 4,3 Data:L9a48/V 5,1 Data:L9a48/V 5,2 Data:L9a48/V 5,3 Data:L9a48/V 5,4 Data:L9a48/V 6,1 Data:L9a48/V 6,2 Data:L9a48/V 6,3 Data:L9a48/V 6,4 Data:L9a48/V 6,5 Data:L9a48/V 6,6 Data:L9a48/V 6,7 Data:L9a48/V 6,8 Data:L9a48/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=$2 is the signature of L9a48. 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 13                   1 -1 11                 2   2 9               2 1   -1 7             4 2     2 5           4 4       0 3         2 2         0 1       3 5           2 -1     1 1             0 -3     3               3 -5 1 1                 0 -7 1                   1

Integral Khovanov Homology (db, data source)

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

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