L6a3

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L6a2

L6a4

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 L6a3's page at Knotilus.

Visit L6a3's page at the original Knot Atlas.

[edit L6a3 Quick Notes]

The link L6a3 is often seen in "Magen David" (star of David) necklaces.

[edit L6a3 Further Notes and Views]

Ruberman, Cochran, Melvin, Akbulut, Gompf, Kirby [1]

Rich Schwartz' "72" [2]

Triangle interlaced with a circle, a traditional symbol of the Christian Trinity (less used in recent centuries)

An architectural trefoil (the outline of three overlapping circles) interlaced with an equilateral triangle, another old Christian Trinitarian symbol.

Further images

Flag of Blonay, Vaud, Switzerland.

Double boomerang like Kolam and Linked knot, Nagata.

[edit] Link Presentations

[edit Notes on L6a3's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_12,5,7,6 X₆₇₁₈ X_4,11,5,12
Gauss code	{1, -2, 3, -6, 4, -5}, {5, -1, 2, -3, 6, -4}

A Braid Representative

A Morse Link Presentation

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{-u^2 v^2-u v-1}{u v}$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}-\frac{1}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z 3 a 7 + 3 z a 7 + a 7 z -1 - z 5 a 5 -5 z 3 a 5 -6 z a 5 - a 5 z -1$ (db)
Kauffman polynomial	$- z a 11 - z 2 a 10 - z 3 a 9 + z a 9 - z 4 a 8 + 2 z 2 a 8 - z 5 a 7 + 4 z 3 a 7 -4 z a 7 + a 7 z -1 - z 4 a 6 + 3 z 2 a 6 - a 6 - z 5 a 5 + 5 z 3 a 5 -6 z a 5 + a 5 z -1$ (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 =

-5 is the signature of L6a3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Data:L6a3/KhovanovTable

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -6$	$i = -4$
$r = -6$	${\mathbb Z}$	${\mathbb Z}$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}$
$r = -2$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -1$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$