L8a4

L8a3

L8a5

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a4 at Knotilus!
	L8a4 is $8^2_{12}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a4's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(t(1)-1) (t(2)-1)^3}{\sqrt{t(1)} t(2)^{3/2}}$ (db)
Jones polynomial	$q^{5/2}-3 q^{3/2}+4 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-z a^5-a^5 z^{-1} +2 z^3 a^3+4 z a^3+3 a^3 z^{-1} -z^5 a-3 z^3 a-4 z a-2 a z^{-1} +z^3 a^{-1} +z a^{-1}$ (db)
Kauffman polynomial	$-a^3 z^7-a z^7-2 a^4 z^6-5 a^2 z^6-3 z^6-2 a^5 z^5-4 a^3 z^5-5 a z^5-3 z^5 a^{-1} -a^6 z^4+5 a^2 z^4-z^4 a^{-2} +3 z^4+3 a^5 z^3+9 a^3 z^3+11 a z^3+5 z^3 a^{-1} +2 a^6 z^2+4 a^4 z^2+2 a^2 z^2+z^2 a^{-2} +z^2-2 a^5 z-7 a^3 z-7 a z-2 z a^{-1} -a^6-3 a^4-3 a^2+a^5 z^{-1} +3 a^3 z^{-1} +2 a z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 χ 6                 1 -1 4               2   2 2             2 1   -1 0           4 2     2 -2         3 4       1 -4       3 2         1 -6     1 3           2 -8   1 3             -2 -10   1               1 -12 1                 -1

Integral Khovanov Homology (db, data source)

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

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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