L4a1

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L2a1

L5a1

1 Link Presentations
2 Polynomial invariants
3 Khovanov Homology
4 Computer Talk
5 Modifying This Page

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L4a1's page at Knotilus.

Visit L4a1's page at the original Knot Atlas.

[edit L4a1 Quick Notes]

L4a1 frequently occurs in late Roman mosaics and some medieval decorations. In this context, it is called the "pelta" or "Solomon's knot" (sigillum Salomonis).

[edit L4a1 Further Notes and Views]

A Kolam with two cycles[1]

Hearst Castle tile [2]

Mosaic seen at Kibbutz Lahav [3]

Carving above door of church in Italy

[edit] Link Presentations

[edit Notes on L4a1's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X₈₃₅₄ X₂₅₃₆ X₄₇₁₈
Gauss code	{1, -3, 2, -4}, {3, -1, 4, -2}

A Braid Representative

A Morse Link Presentation

[edit] Polynomial invariants

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

[edit] Khovanov Homology

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

-1 is the signature of L4a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Data:L4a1/KhovanovTable

Integral Khovanov Homology

(db, data source)

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