L8n7

L8n6

L8n8

	Visit L8n7's page at Knotilus! Visit L8n7's page at the original Knot Atlas!
	L8n7 is $8_{2}^{4}$ in the Rolfsen table of links.

L8n7 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X2536 X11,13,12,16 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X15,9,16,12
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 8}, {-7, 6, -8, 3}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)-t(1)t(3)t(2)+t(3)t(2)-t(1)t(4)t(2)-t(3)+t(1)t(4)+t(3)t(4)-t(4)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}{\sqrt {t(4)}}}}$ (db)
Jones polynomial	$-q^{13/2}+q^{11/2}-4q^{9/2}+q^{7/2}-4q^{5/2}+2q^{3/2}-3{\sqrt {q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-a^{-7}z^{-3}-a^{-7}z^{-1}+3a^{-5}z^{-3}+3za^{-5}+5a^{-5}z^{-1}-2z^{3}a^{-3}-3a^{-3}z^{-3}-6za^{-3}-7a^{-3}z^{-1}+a^{-1}z^{-3}+3za^{-1}+3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-4z^{3}a^{-7}+a^{-7}z^{-3}+6za^{-7}-4a^{-7}z^{-1}+z^{6}a^{-6}-z^{4}a^{-6}-6z^{2}a^{-6}-3a^{-6}z^{-2}+8a^{-6}+5z^{5}a^{-5}-16z^{3}a^{-5}+3a^{-5}z^{-3}+14za^{-5}-9a^{-5}z^{-1}+z^{6}a^{-4}+2z^{4}a^{-4}-12z^{2}a^{-4}-6a^{-4}z^{-2}+15a^{-4}+4z^{5}a^{-3}-12z^{3}a^{-3}+3a^{-3}z^{-3}+14za^{-3}-9a^{-3}z^{-1}+3z^{4}a^{-2}-6z^{2}a^{-2}-3a^{-2}z^{-2}+8a^{-2}+a^{-1}z^{-3}+6za^{-1}-4a^{-1}z^{-1}$ (db)

Vassiliev invariants

V2 and V3:

(0, ${\frac {77}{6}}$)

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:L8n7/V 2,1 Data:L8n7/V 3,1 Data:L8n7/V 4,1 Data:L8n7/V 4,2 Data:L8n7/V 4,3 Data:L8n7/V 5,1 Data:L8n7/V 5,2 Data:L8n7/V 5,3 Data:L8n7/V 5,4 Data:L8n7/V 6,1 Data:L8n7/V 6,2 Data:L8n7/V 6,3 Data:L8n7/V 6,4 Data:L8n7/V 6,5 Data:L8n7/V 6,6 Data:L8n7/V 6,7 Data:L8n7/V 6,8 Data:L8n7/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 L8n7. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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[8, NonAlternating, 7]]
Out[2]=	8
In[3]:=	PD[Link[8, NonAlternating, 7]]
Out[3]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[11, 13, 12, 16], X[3, 11, 4, 10], X[9, 1, 10, 4], X[7, 15, 8, 14], X[13, 5, 14, 8], X[15, 9, 16, 12]]
In[4]:=	GaussCode[Link[8, NonAlternating, 7]]
Out[4]=	GaussCode[{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 8}, {-7, 6, -8, 3}]
In[5]:=	BR[Link[8, NonAlternating, 7]]
Out[5]=	BR[Link[8, NonAlternating, 7]]
In[6]:=	alex = Alexander[Link[8, NonAlternating, 7]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, NonAlternating, 7]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, NonAlternating, 7]], KnotSignature[Link[8, NonAlternating, 7]]}
Out[9]=	{Infinity, 1}
In[10]:=	J=Jones[Link[8, NonAlternating, 7]][q]
Out[10]=	3/2 5/2 7/2 9/2 11/2 13/2 -3 Sqrt[q] + 2 q - 4 q + q - 4 q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[8, NonAlternating, 7]][q]
Out[12]=	2 4 6 8 10 12 14 16 3 + 3 q + 5 q + 8 q + 10 q + 13 q + 12 q + 11 q + 8 q + 18 20 22 4 q + 3 q + q
In[13]:=	Kauffman[Link[8, NonAlternating, 7]][a, z]
Out[13]=	-8 15 8 1 3 3 1 3 6 3 -- - -- - -- - ----- - ----- - ----- - ---- + ----- + ----- + ----- + 6 4 2 7 3 5 3 3 3 3 6 2 4 2 2 2 a a a a z a z a z a z a z a z a z 2 2 4 9 9 4 6 z 14 z 14 z 6 z 6 z 12 z ---- + ---- + ---- + --- - --- - ---- - ---- - --- + ---- + ----- + 7 5 3 a z 7 5 3 a 6 4 a z a z a z a a a a a 2 3 3 3 4 4 4 5 5 5 6 z 4 z 16 z 12 z z 2 z 3 z z 5 z 4 z ---- + ---- + ----- + ----- + -- - ---- - ---- - -- - ---- - ---- - 2 7 5 3 6 4 2 7 5 3 a a a a a a a a a a 6 6 z z -- - -- 6 4 a a
In[14]:=	{Vassiliev[2][Link[8, NonAlternating, 7]], Vassiliev[3][Link[8, NonAlternating, 7]]}
Out[14]=	77 {0, --} 6
In[15]:=	Kh[Link[8, NonAlternating, 7]][q, t]
Out[15]=	2 4 2 4 2 6 2 8 3 8 4 10 4 3 + 4 q + q + 3 q t + q t + 3 q t + q t + 4 q t + 4 q t + 10 5 14 6 q t + q t