L7n2

(Knotscape image)

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

Visit L7n2's page at the original Knot Atlas.

L7n2 is $7^2_8$ in the Rolfsen table of links.

Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_13,1,14,4 X_5,10,6,11 X₃₈₄₉ X_9,14,10,5 X_2,12,3,11
Gauss code	{1, -7, -5, 3}, {-4, -1, 2, 5, -6, 4, 7, -2, -3, 6}

A Braid Representative

A Morse Link Presentation

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

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 L7n2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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