L9a28

L9a27

L9a29

	Visit L9a28's page at Knotilus! Visit L9a28's page at the original Knot Atlas!
	L9a28 is $9_{20}^{2}$ in the Rolfsen table of links.

L9a28 Further Notes and Views

Knot presentations

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X18,16,7,15 X16,12,17,11 X12,18,13,17 X6718 X4,13,5,14
Gauss code	{1, -2, 3, -9, 4, -8}, {8, -1, 2, -3, 6, -7, 9, -4, 5, -6, 7, -5}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(2)^{4}-2t(1)^{2}t(2)^{3}+2t(1)t(2)^{3}+2t(1)^{2}t(2)^{2}-3t(1)t(2)^{2}+2t(2)^{2}+2t(1)t(2)-2t(2)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	${\frac {5}{q^{9/2}}}-{\frac {6}{q^{7/2}}}+{\frac {5}{q^{5/2}}}+q^{3/2}-{\frac {5}{q^{3/2}}}-{\frac {1}{q^{15/2}}}+{\frac {2}{q^{13/2}}}-{\frac {4}{q^{11/2}}}-2{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{5}z^{5}+4a^{5}z^{3}+5a^{5}z+2a^{5}z^{-1}-a^{3}z^{7}-6a^{3}z^{5}-13a^{3}z^{3}-12a^{3}z-3a^{3}z^{-1}+az^{5}+4az^{3}+4az+az^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{3}-a^{9}z+2a^{8}z^{4}-a^{8}z^{2}+3a^{7}z^{5}-3a^{7}z^{3}+2a^{7}z+3a^{6}z^{6}-3a^{6}z^{4}+a^{6}z^{2}+3a^{5}z^{7}-7a^{5}z^{5}+9a^{5}z^{3}-7a^{5}z+2a^{5}z^{-1}+a^{4}z^{8}+2a^{4}z^{6}-11a^{4}z^{4}+10a^{4}z^{2}-3a^{4}+5a^{3}z^{7}-18a^{3}z^{5}+22a^{3}z^{3}-14a^{3}z+3a^{3}z^{-1}+a^{2}z^{8}-10a^{2}z^{4}+12a^{2}z^{2}-3a^{2}+2az^{7}-8az^{5}+9az^{3}-4az+az^{-1}+z^{6}-4z^{4}+4z^{2}-1$ (db)

Vassiliev invariants

V2 and V3:

(0, $-{\frac {121}{16}}$)

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:L9a28/V 2,1 Data:L9a28/V 3,1 Data:L9a28/V 4,1 Data:L9a28/V 4,2 Data:L9a28/V 4,3 Data:L9a28/V 5,1 Data:L9a28/V 5,2 Data:L9a28/V 5,3 Data:L9a28/V 5,4 Data:L9a28/V 6,1 Data:L9a28/V 6,2 Data:L9a28/V 6,3 Data:L9a28/V 6,4 Data:L9a28/V 6,5 Data:L9a28/V 6,6 Data:L9a28/V 6,7 Data:L9a28/V 6,8 Data:L9a28/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 L9a28. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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, 28]]
Out[2]=	9
In[3]:=	PD[Link[9, Alternating, 28]]
Out[3]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[14, 5, 15, 6], X[18, 16, 7, 15], X[16, 12, 17, 11], X[12, 18, 13, 17], X[6, 7, 1, 8], X[4, 13, 5, 14]]
In[4]:=	GaussCode[Link[9, Alternating, 28]]
Out[4]=	GaussCode[{1, -2, 3, -9, 4, -8}, {8, -1, 2, -3, 6, -7, 9, -4, 5, -6, 7, -5}]
In[5]:=	BR[Link[9, Alternating, 28]]
Out[5]=	BR[Link[9, Alternating, 28]]
In[6]:=	alex = Alexander[Link[9, Alternating, 28]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[9, Alternating, 28]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[9, Alternating, 28]], KnotSignature[Link[9, Alternating, 28]]}
Out[9]=	{Infinity, -3}
In[10]:=	J=Jones[Link[9, Alternating, 28]][q]
Out[10]=	-(15/2) 2 4 5 6 5 5 3 -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, 28]][q]
Out[12]=	-22 2 -16 -14 2 3 -4 4 -1 + q + --- + q + q + --- + -- + q - q 18 12 8 q q q
In[13]:=	Kauffman[Link[9, Alternating, 28]][a, z]
Out[13]=	3 5 2 4 a 3 a 2 a 3 5 7 1 + 3 a + 3 a - - - ---- - ---- + 4 a z + 14 a z + 7 a z - 2 a z + z z z 9 2 2 2 4 2 6 2 8 2 3 a z - 4 z - 12 a z - 10 a z - a z + a z - 9 a z - 3 3 5 3 7 3 9 3 4 2 4 4 4 22 a z - 9 a z + 3 a z - a z + 4 z + 10 a z + 11 a z + 6 4 8 4 5 3 5 5 5 7 5 6 3 a z - 2 a z + 8 a z + 18 a z + 7 a z - 3 a z - z - 4 6 6 6 7 3 7 5 7 2 8 4 8 2 a z - 3 a z - 2 a z - 5 a z - 3 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[9, Alternating, 28]], Vassiliev[3][Link[9, Alternating, 28]]}
Out[14]=	121 {0, -(---)} 16
In[15]:=	Kh[Link[9, Alternating, 28]][q, t]
Out[15]=	3 3 1 1 2 2 2 3 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 4 2 16 6 14 6 14 5 12 4 10 4 10 3 8 3 q q q t q t q t q t q t q t q t 3 3 2 3 t 2 2 2 4 3 ----- + ----- + ---- + ---- + 2 t + -- + t + q t + q t 8 2 6 2 6 4 2 q t q t q t q t q