L11n133

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_18,11,19,12 X_3,10,4,11 X_2,17,3,18 X_12,5,13,6 X₆₇₁₈ X_9,16,10,17 X_13,20,14,21 X_15,22,16,7 X_19,4,20,5 X_21,14,22,15
Gauss code	{1, -4, -3, 10, 5, -6}, {6, -1, -7, 3, 2, -5, -8, 11, -9, 7, 4, -2, -10, 8, -11, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 6 - v u 4 + v u 3 - v u 2 -1$ (db)
Jones polynomial	$-\frac{1}{q^{9/2}}-\frac{1}{q^{13/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{21/2}}-\frac{1}{q^{23/2}}+\frac{1}{q^{25/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$- z 3 a 13 -3 z a 13 -2 a 13 z -1 + z 7 a 11 + 8 z 5 a 11 + 20 z 3 a 11 + 18 z a 11 + 5 a 11 z -1 - z 9 a 9 -9 z 7 a 9 -28 z 5 a 9 -36 z 3 a 9 -18 z a 9 -3 a 9 z -1$ (db)
Kauffman polynomial	$- z 2 a 16 + a 16 - z 3 a 15 + z a 15 - z 2 a 14 + z 3 a 13 -4 z a 13 + 2 a 13 z -1 - z 8 a 12 + 8 z 6 a 12 -20 z 4 a 12 + 18 z 2 a 12 -5 a 12 - z 9 a 11 + 9 z 7 a 11 -28 z 5 a 11 + 38 z 3 a 11 -23 z a 11 + 5 a 11 z -1 - z 8 a 10 + 8 z 6 a 10 -20 z 4 a 10 + 18 z 2 a 10 -5 a 10 - z 9 a 9 + 9 z 7 a 9 -28 z 5 a 9 + 36 z 3 a 9 -18 z a 9 + 3 a 9 z -1$ (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 =

-7 is the signature of L11n133. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -10$	$i = -8$	$i = -6$
$r = -9$		${\mathbb Z}$
$r = -8$		${\mathbb Z}_2$	${\mathbb Z}$
$r = -7$		${\mathbb Z}$
$r = -6$		${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = -5$	${\mathbb Z}$	${\mathbb Z}$
$r = -4$		${\mathbb Z}$	${\mathbb Z}$
$r = -3$	${\mathbb Z}$
$r = -2$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -1$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$