L8a17

L8a16

L8a18

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a17 at Knotilus!
	L8a17 is [math]\displaystyle{ 8^3_{2} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a17's Link Presentations]

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

A Braid Representative
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(3)^2 t(2)^2+t(1) t(3) t(2)^2-t(3) t(2)^2+t(1) t(3)^2 t(2)-t(3)^2 t(2)+t(1) t(2)-2 t(1) t(3) t(2)+2 t(3) t(2)-t(2)-t(1)+t(1) t(3)-t(3)}{\sqrt{t(1)} t(2) t(3)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-10} -2 q^{-9} +4 q^{-8} -4 q^{-7} +6 q^{-6} -4 q^{-5} +4 q^{-4} -2 q^{-3} + q^{-2} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{10} z^{-2} +a^{10}-3 a^8 z^2-2 a^8 z^{-2} -6 a^8+2 a^6 z^4+6 a^6 z^2+a^6 z^{-2} +5 a^6+a^4 z^4+2 a^4 z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{12} z^4-2 a^{12} z^2+a^{12}+2 a^{11} z^5-3 a^{11} z^3+2 a^{10} z^6-2 a^{10} z^4+a^{10} z^2+a^{10} z^{-2} -3 a^{10}+a^9 z^7+2 a^9 z^5-6 a^9 z^3+6 a^9 z-2 a^9 z^{-1} +5 a^8 z^6-12 a^8 z^4+15 a^8 z^2+2 a^8 z^{-2} -8 a^8+a^7 z^7+2 a^7 z^5-6 a^7 z^3+6 a^7 z-2 a^7 z^{-1} +3 a^6 z^6-8 a^6 z^4+10 a^6 z^2+a^6 z^{-2} -5 a^6+2 a^5 z^5-3 a^5 z^3+a^4 z^4-2 a^4 z^2 }[/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 -3                 1 1 -5               2 1 -1 -7             2     2 -9           2 2     0 -11         4 2       2 -13       1 3         2 -15     3 3           0 -17   1 3             2 -19   1               -1 -21 1                 1

Integral Khovanov Homology (db, data source)

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

L8a16 L8a18