L11n42

(Knotscape image)

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

Visit L11n42's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_9,18,10,19 X₃₈₄₉ X_5,14,6,15 X_15,22,16,5 X_17,20,18,21 X_21,16,22,17 X_19,10,20,11 X_11,2,12,3
Gauss code	{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 10, -11, -2, 3, 6, -7, 9, -8, 4, -10, 8, -9, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-3 v u 3 + 4 v u 2 -2 u 2 -2 v u + 4 u -3$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{6}{q^{13/2}}+\frac{6}{q^{15/2}}-\frac{6}{q^{17/2}}+\frac{4}{q^{19/2}}-\frac{3}{q^{21/2}}+\frac{2}{q^{23/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- a 13 z -1 + z 3 a 11 + 3 z a 11 + 2 a 11 z -1 - z 5 a 9 -3 z 3 a 9 -2 z a 9 - a 9 z -1 - z 5 a 7 -2 z 3 a 7 + z a 7 + a 7 z -1 - z 5 a 5 -4 z 3 a 5 -4 z a 5 - a 5 z -1$ (db)
Kauffman polynomial	$-3 z 4 a 14 + 8 z 2 a 14 -3 a 14 - z 7 a 13 + z 5 a 13 + 3 z 3 a 13 -3 z a 13 + a 13 z -1 -2 z 8 a 12 + 10 z 6 a 12 -24 z 4 a 12 + 24 z 2 a 12 -7 a 12 - z 9 a 11 + 4 z 7 a 11 -10 z 5 a 11 + 11 z 3 a 11 -8 z a 11 + 2 a 11 z -1 -3 z 8 a 10 + 13 z 6 a 10 -25 z 4 a 10 + 16 z 2 a 10 -4 a 10 - z 9 a 9 + 4 z 7 a 9 -9 z 5 a 9 + 7 z 3 a 9 -4 z a 9 + a 9 z -1 - z 8 a 8 + 2 z 6 a 8 -2 z 4 a 8 + z 2 a 8 - z 7 a 7 + z 5 a 7 + 3 z 3 a 7 -3 z a 7 + a 7 z -1 - z 6 a 6 + 2 z 4 a 6 + z 2 a 6 - a 6 - z 5 a 5 + 4 z 3 a 5 -4 z a 5 + a 5 z -1$ (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 =

-5 is the signature of L11n42. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -6$	$i = -4$
$r = -9$	${\mathbb Z}^{2}$
$r = -8$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -7$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{4}$
$r = -3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -2$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -1$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$