L8a18

L8a17

L8a19

	Visit L8a18's page at Knotilus! Visit L8a18's page at the original Knot Atlas!
	L8a18 is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8^3_{1}} in the Rolfsen table of links.

L8a18 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w} , ...)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{u v^2 w^2-u v w^2+u v w-u w+u-v^2 w^2+v^2 w-v w+v-1}{\sqrt{u} v w}} (db)
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^8-2 q^7+3 q^6-3 q^5+4 q^4-2 q^3+3 q^2-q+1} (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{4}a^{-6}+3z^{2}a^{-6}+a^{-6}z^{-2}+2a^{-6}-z^{6}a^{-4}-5z^{4}a^{-4}-8z^{2}a^{-4}-2a^{-4}z^{-2}-6a^{-4}+z^{4}a^{-2}+4z^{2}a^{-2}+a^{-2}z^{-2}+4a^{-2}$ (db)
Kauffman polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^2 a^{-10} +2 z^3 a^{-9} +3 z^4 a^{-8} -3 z^2 a^{-8} + a^{-8} +3 z^5 a^{-7} -4 z^3 a^{-7} +3 z^6 a^{-6} -8 z^4 a^{-6} +6 z^2 a^{-6} + a^{-6} z^{-2} -3 a^{-6} +z^7 a^{-5} -7 z^3 a^{-5} +6 z a^{-5} -2 a^{-5} z^{-1} +4 z^6 a^{-4} -16 z^4 a^{-4} +18 z^2 a^{-4} +2 a^{-4} z^{-2} -8 a^{-4} +z^7 a^{-3} -3 z^5 a^{-3} -z^3 a^{-3} +6 z a^{-3} -2 a^{-3} z^{-1} +z^6 a^{-2} -5 z^4 a^{-2} +8 z^2 a^{-2} + a^{-2} z^{-2} -5 a^{-2} } (db)

Vassiliev invariants

V2 and V3:

(0, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{26}{3}} )

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:L8a18/V 2,1 Data:L8a18/V 3,1 Data:L8a18/V 4,1 Data:L8a18/V 4,2 Data:L8a18/V 4,3 Data:L8a18/V 5,1 Data:L8a18/V 5,2 Data:L8a18/V 5,3 Data:L8a18/V 5,4 Data:L8a18/V 6,1 Data:L8a18/V 6,2 Data:L8a18/V 6,3 Data:L8a18/V 6,4 Data:L8a18/V 6,5 Data:L8a18/V 6,6 Data:L8a18/V 6,7 Data:L8a18/V 6,8 Data:L8a18/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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or $j-2r=s-1$, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 4 is the signature of L8a18. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 χ 17                 1 1 15               2 1 -1 13             1     1 11           2 2     0 9         2 1       1 7       1 3         2 5     2 1           1 3   1 3             2 1                   0 -1 1                 1

Integral Khovanov Homology (db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} $r=-1$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} $r=1$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{2}} ${\mathbb {Z} }^{2}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=6} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{Include}(\textrm{ColouredJonesM.mhtml})}

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Link[8, Alternating, 18]]
Out[2]=	8
In[3]:=	PD[Link[8, Alternating, 18]]
Out[3]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[8, 16, 9, 15], X[14, 8, 15, 7], X[16, 10, 11, 9], X[10, 12, 5, 11], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[4]:=	GaussCode[Link[8, Alternating, 18]]
Out[4]=	GaussCode[{1, -7, 2, -8}, {7, -1, 4, -3, 5, -6}, {6, -2, 8, -4, 3, -5}]
In[5]:=	BR[Link[8, Alternating, 18]]
Out[5]=	BR[Link[8, Alternating, 18]]
In[6]:=	alex = Alexander[Link[8, Alternating, 18]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, Alternating, 18]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, Alternating, 18]], KnotSignature[Link[8, Alternating, 18]]}
Out[9]=	{Infinity, 4}
In[10]:=	J=Jones[Link[8, Alternating, 18]][q]
Out[10]=	2 3 4 5 6 7 8 1 - q + 3 q - 2 q + 4 q - 3 q + 3 q - 2 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[8, Alternating, 18]][q]
Out[12]=	2 4 6 8 10 12 14 16 18 1 + q + 2 q + 3 q + 3 q + 5 q + 3 q + 4 q + 2 q + q + 20 24 q + q
In[13]:=	Kauffman[Link[8, Alternating, 18]][a, z]
Out[13]=	-8 3 8 5 1 2 1 2 2 6 z 6 z a - -- - -- - -- + ----- + ----- + ----- - ---- - ---- + --- + --- + 6 4 2 6 2 4 2 2 2 5 3 5 3 a a a a z a z a z a z a z a a 2 2 2 2 2 3 3 3 3 4 z 3 z 6 z 18 z 8 z 2 z 4 z 7 z z 3 z --- - ---- + ---- + ----- + ---- + ---- - ---- - ---- - -- + ---- - 10 8 6 4 2 9 7 5 3 8 a a a a a a a a a a 4 4 4 5 5 6 6 6 7 7 8 z 16 z 5 z 3 z 3 z 3 z 4 z z z z ---- - ----- - ---- + ---- - ---- + ---- + ---- + -- + -- + -- 6 4 2 7 3 6 4 2 5 3 a a a a a a a a a a
In[14]:=	{Vassiliev[2][Link[8, Alternating, 18]], Vassiliev[3][Link[8, Alternating, 18]]}
Out[14]=	26 {0, -(--)} 3
In[15]:=	Kh[Link[8, Alternating, 18]][q, t]
Out[15]=	3 3 5 1 q 5 7 7 2 9 2 9 3 3 q + 2 q + ---- + -- + q t + q t + 3 q t + 2 q t + q t + 2 t q t 11 3 11 4 13 4 15 5 15 6 17 6 2 q t + 2 q t + q t + 2 q t + q t + q t