L8a13

L8a12

L8a14

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a13 at Knotilus!
	L8a13 is [math]\displaystyle{ 8^2_{4} }[/math] in the Rolfsen table of links. Contains two L4a1 configurations.

Symmetric form

(alternate)

National swiss museum

Povray depiction

Link Presentations

[edit Notes on L8a13's Link Presentations]

Planar diagram presentation	X10,1,11,2 X16,7,9,8 X12,3,13,4 X6,13,7,14 X14,5,15,6 X4,15,5,16 X2,9,3,10 X8,11,1,12
Gauss code	{1, -7, 3, -6, 5, -4, 2, -8}, {7, -1, 8, -3, 4, -5, 6, -2}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ -\frac{(t(1)+t(2)) \left(t(2) t(1)^2+t(2)^2 t(1)-2 t(2) t(1)+t(1)+t(2)\right)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{4}{q^{9/2}}-\frac{4}{q^{7/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{4}{q^{13/2}}-\frac{4}{q^{11/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^9+a^9 z^{-1} -z^3 a^7-z a^7-a^7 z^{-1} -2 z^3 a^5-3 z a^5-z^3 a^3-z a^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{11}+4 z^3 a^{11}-4 z a^{11}-z^6 a^{10}+2 z^4 a^{10}-z^7 a^9+2 z^5 a^9-2 z^3 a^9+3 z a^9-a^9 z^{-1} -3 z^6 a^8+5 z^4 a^8-z^2 a^8+a^8-z^7 a^7-z^3 a^7+3 z a^7-a^7 z^{-1} -2 z^6 a^6+z^4 a^6-3 z^5 a^5+4 z^3 a^5-3 z a^5-2 z^4 a^4+z^2 a^4-z^3 a^3+z a^3 }[/math] (db)

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]).

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                 1 1 -4               2 1 -1 -6             2     2 -8           2 2     0 -10         2 2       0 -12       2 2         0 -14     1 2           -1 -16     2             2 -18 1 1               0 -20 1                 1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

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

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

L8a12 L8a14