L11n354

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,18,12,19 X_7,16,8,17 X_15,8,16,9 X_13,21,14,20 X_19,22,20,15 X_21,13,22,12 X_17,14,18,5 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, -9, 3, -7, 6, -8, 7}, {10, -1, -4, 5, 11, -2, -3, 8, -6, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 4 + u 4 + 2 v u 3 - v 2 w u 3 + v w u 3 - u 3 + v 2 w u 2 - u 2 - v u + v 2 w u -2 v w u + u - v 2 w + v w$ (db)
Jones polynomial	$1- q -1 + q -2 + 2 q -5 - q -6 + 3 q -7 -2 q -8 + 2 q -9 - q -10$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 10 + 2 z 2 a 8 + a 8 z -2 + 2 a 8 + z 2 a 6 -2 a 6 z -2 -2 a 6 - z 6 a 4 -5 z 4 a 4 -5 z 2 a 4 + a 4 z -2 - a 4 + z 4 a 2 + 4 z 2 a 2 + 2 a 2$ (db)
Kauffman polynomial	$z 7 a 11 -5 z 5 a 11 + 6 z 3 a 11 -2 z a 11 + 2 z 8 a 10 -11 z 6 a 10 + 17 z 4 a 10 -11 z 2 a 10 + 3 a 10 + z 9 a 9 -3 z 7 a 9 -6 z 5 a 9 + 16 z 3 a 9 -7 z a 9 + 4 z 8 a 8 -25 z 6 a 8 + 46 z 4 a 8 -35 z 2 a 8 - a 8 z -2 + 12 a 8 + z 9 a 7 -4 z 7 a 7 -4 z 5 a 7 + 19 z 3 a 7 -14 z a 7 + 2 a 7 z -1 + 2 z 8 a 6 -14 z 6 a 6 + 28 z 4 a 6 -26 z 2 a 6 -2 a 6 z -2 + 12 a 6 + z 7 a 5 -8 z 5 a 5 + 14 z 3 a 5 -10 z a 5 + 2 a 5 z -1 + z 6 a 4 -6 z 4 a 4 + 4 z 2 a 4 - a 4 z -2 + 2 a 4 + z 7 a 3 -5 z 5 a 3 + 5 z 3 a 3 - z a 3 + z 6 a 2 -5 z 4 a 2 + 6 z 2 a 2 -2 a 2$ (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 L11n354. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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