L7a5

L7a4

L7a6

	Visit L7a5's page at Knotilus! Visit L7a5's page at the original Knot Atlas!
	L7a5 is $7_{2}^{2}$ in the Rolfsen table of links.

L7a5 Further Notes and Views

Knot presentations

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

Polynomial invariants

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

Vassiliev invariants

V2 and V3:

(0, ${\frac {31}{48}}$)

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:L7a5/V 2,1 Data:L7a5/V 3,1 Data:L7a5/V 4,1 Data:L7a5/V 4,2 Data:L7a5/V 4,3 Data:L7a5/V 5,1 Data:L7a5/V 5,2 Data:L7a5/V 5,3 Data:L7a5/V 5,4 Data:L7a5/V 6,1 Data:L7a5/V 6,2 Data:L7a5/V 6,3 Data:L7a5/V 6,4 Data:L7a5/V 6,5 Data:L7a5/V 6,6 Data:L7a5/V 6,7 Data:L7a5/V 6,8 Data:L7a5/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 L7a5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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[7, Alternating, 5]]
Out[2]=	7
In[3]:=	PD[Link[7, Alternating, 5]]
Out[3]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[12, 6, 13, 5], X[14, 11, 7, 12], X[4, 14, 5, 13], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[4]:=	GaussCode[Link[7, Alternating, 5]]
Out[4]=	GaussCode[{1, -6, 2, -5, 3, -7}, {6, -1, 7, -2, 4, -3, 5, -4}]
In[5]:=	BR[Link[7, Alternating, 5]]
Out[5]=	BR[Link[7, Alternating, 5]]
In[6]:=	alex = Alexander[Link[7, Alternating, 5]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[7, Alternating, 5]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[7, Alternating, 5]], KnotSignature[Link[7, Alternating, 5]]}
Out[9]=	{Infinity, -1}
In[10]:=	J=Jones[Link[7, Alternating, 5]][q]
Out[10]=	-(11/2) 2 2 4 3 3 3/2 q - ---- + ---- - ---- + ---- - ------- + 2 Sqrt[q] - q 9/2 7/2 5/2 3/2 Sqrt[q] q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[7, Alternating, 5]][q]
Out[12]=	-18 -14 2 2 2 2 2 6 1 - q + q + --- + -- + -- + -- - q + q 10 8 6 4 q q q q
In[13]:=	Kauffman[Link[7, Alternating, 5]][a, z]
Out[13]=	3 2 a a z 3 5 2 2 2 4 2 -a + - + -- + - - 2 a z - 6 a z - 3 a z + 2 z + 2 a z + 2 a z + z z a 3 6 2 z 3 3 3 5 3 4 2 4 6 4 2 a z - -- + 2 a z + 8 a z + 5 a z - 2 z - a z - a z - a 5 3 5 5 5 2 6 4 6 2 a z - 4 a z - 2 a z - a z - a z
In[14]:=	{Vassiliev[2][Link[7, Alternating, 5]], Vassiliev[3][Link[7, Alternating, 5]]}
Out[14]=	31 {0, --} 48
In[15]:=	Kh[Link[7, Alternating, 5]][q, t]
Out[15]=	2 1 1 1 1 1 3 2 2 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 q q t q t q t q t q t q t q t 1 2 2 4 2 ---- + ---- + t + q t + q t 4 2 q t q t