L8n2

L8n1

L8n3

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n2 at Knotilus!
	L8n2 is $8^2_{15}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8n2's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X15,1,16,4 X9,12,10,13 X3849 X5,11,6,10 X11,5,12,16 X2,14,3,13
Gauss code	{1, -8, -5, 3}, {-6, -1, 2, 5, -4, 6, -7, 4, 8, -2, -3, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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