L8a19

L8a18

L8a20

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a19 at Knotilus!
	L8a19 is $8^3_{6}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a19's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,5,15,6 X8,12,9,11 X16,8,11,7 X10,13,5,14 X2,9,3,10 X4,16,1,15
Gauss code	{1, -7, 2, -8}, {3, -1, 5, -4, 7, -6}, {4, -2, 6, -3, 8, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(t(1)-1) (t(3) t(2)-t(2)+1) (t(2) t(3)-t(3)+1)}{\sqrt{t(1)} t(2) t(3)}$ (db)
Jones polynomial	$q^4+ q^{-4} -3 q^3-3 q^{-3} +5 q^2+5 q^{-2} -5 q-5 q^{-1} +8$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^6+a^2 z^4+z^4 a^{-2} -4 z^4+2 a^2 z^2+2 z^2 a^{-2} -5 z^2+a^2+ a^{-2} -2+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2}$ (db)
Kauffman polynomial	$2 a z^7+2 z^7 a^{-1} +4 a^2 z^6+4 z^6 a^{-2} +8 z^6+3 a^3 z^5+a z^5+z^5 a^{-1} +3 z^5 a^{-3} +a^4 z^4-7 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} -16 z^4-4 a^3 z^3-4 a z^3-4 z^3 a^{-1} -4 z^3 a^{-3} -a^4 z^2+5 a^2 z^2+5 z^2 a^{-2} -z^2 a^{-4} +12 z^2+2 a z+2 z a^{-1} -2 a^2-2 a^{-2} -3-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 χ 9                 1 1 7               2   -2 5             3 1   2 3           3 3     0 1         5 2       3 -1       2 5         3 -3     3 3           0 -5   1 3             2 -7   2               -2 -9 1                 1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

Computer Talk

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

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.

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