L11n349

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_20,12,21,11 X_18,8,19,7 X_22,18,13,17 X_9,17,10,16 X_15,11,16,10 X_12,20,5,19 X_8,22,9,21 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, -6, 7, 3, -8}, {11, -2, -7, 6, 5, -4, 8, -3, 9, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v 2 u 3 - v u 3 - v 2 w u 3 + v w u 3 + v 3 u 2 -3 v 2 u 2 + 2 v u 2 + 2 v 2 w u 2 -2 v w u 2 + 2 v 2 u -2 v u -2 v 2 w u + 3 v w u - w u - v 2 + v + v 2 w -2 v w$ (db)
Jones polynomial	$- q 6 + 5 q 5 -7 q 4 + 10 q 3 -10 q 2 + 10 q -8 + 6 q -1 -2 q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + 2 z 4 a -2 - z 4 a -4 -2 z 4 + a 2 z 2 -2 z 2 a -2 -4 z 2 + 2 a 2 -5 a -2 + 3 a -4 -2 a -2 z -2 + a -4 z -2 + z -2$ (db)
Kauffman polynomial	$z 9 a -1 + z 9 a -3 + 6 z 8 a -2 + 3 z 8 a -4 + 3 z 8 + 2 a z 7 + 5 z 7 a -1 + 5 z 7 a -3 + 2 z 7 a -5 + a 2 z 6 -14 z 6 a -2 -5 z 6 a -4 -8 z 6 -5 a z 5 -20 z 5 a -1 -15 z 5 a -3 -4 a 2 z 4 + 14 z 4 a -2 + 8 z 4 a -4 + 5 z 4 a -6 + 7 z 4 + 2 a z 3 + 17 z 3 a -1 + 15 z 3 a -3 + z 3 a -5 + z 3 a -7 + 5 a 2 z 2 -16 z 2 a -2 -6 z 2 a -4 - z 2 a -6 -6 z 2 + a z -8 z a -1 -8 z a -3 + z a -5 -2 a 2 + 9 a -2 + 3 a -4 -2 a -6 + 3 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 =

2 is the signature of L11n349. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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