L5a1

L4a1

L6a1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L5a1 at Knotilus!
	L5a1 is $5^2_1$ in Rolfsen's Table of Links. It is also known as the "Whitehead Link".

Basic depiction		Drawing of "Thor's hammer" or Mjölnir found in Sweden		Wolfgang Staubach's medallion based on this [1]
A kolam with two cycles, one of which is twisted[2]		A simplest closed-loop version of heraldic "fret" / "fretty" ornamentation.		Bisexuality symbol.

Link Presentations

[edit Notes on L5a1's Link Presentations]

Planar diagram presentation	X6172 X10,7,5,8 X4516 X2,10,3,9 X8493
Gauss code	{1, -4, 5, -3}, {3, -1, 2, -5, 4, -2}

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	$\frac{1}{q^{7/2}}-\frac{2}{q^{5/2}}-q^{3/2}+\frac{1}{q^{3/2}}+\sqrt{q}-\frac{2}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-z a^3+z^3 a+2 z a+a z^{-1} -z a^{-1} - a^{-1} z^{-1}$ (db)
Kauffman polynomial	$-z^2 a^4-2 z^3 a^3+2 z a^3-z^4 a^2-3 z^3 a+4 z a-a z^{-1} -z^4+z^2+1-z^3 a^{-1} +2 z a^{-1} - a^{-1} z^{-1}$ (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   \ -3 -2 -1 0 1 2 χ 4           1 1 2             0 0       2 1   1 -2     1 2     1 -4   1         1 -6   1         1 -8 1           -1

Integral Khovanov Homology (db, data source)

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

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.

L4a1 L6a1