L5a1

L4a1

L6a1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L5a1 at Knotilus!
	L5a1 is $5_{1}^{2}$ 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	$-za^{3}+z^{3}a+2za+az^{-1}-za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{2}a^{4}-2z^{3}a^{3}+2za^{3}-z^{4}a^{2}-3z^{3}a+4za-az^{-1}-z^{4}+z^{2}+1-z^{3}a^{-1}+2za^{-1}-a^{-1}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ 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)

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

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