L11n372

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_3,10,4,11 X_7,19,8,18 X_15,17,16,22 X_9,20,10,21 X_13,9,14,8 X_17,15,18,14 X_21,5,22,16 X_19,12,20,13 X₂₅₃₆ X_11,4,12,1
Gauss code	{1, -10, -2, 11}, {-7, 3, -9, 5, -8, 4}, {10, -1, -3, 6, -5, 2, -11, 9, -6, 7, -4, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v w u 4 + w u 4 - v 2 w u 3 + 3 v w u 3 -2 w u 3 + u 3 - v 2 u 2 + 2 v u 2 + 2 v 2 w u 2 -2 v w u 2 + w u 2 -2 u 2 + 2 v 2 u -3 v u - v 2 w u + u - v 2 + v$ (db)
Jones polynomial	$- q 4 + 3 q 3 -5 q 2 + 8 q -8 + 10 q -1 -8 q -2 + 7 q -3 -4 q -4 + 2 q -5$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a 6 - z 4 a 4 -3 z 2 a 4 + a 4 z -2 - a 4 + z 6 a 2 + 3 z 4 a 2 + z 2 a 2 -2 a 2 z -2 -2 a 2 + z 6 + 3 z 4 + 2 z 2 + z -2 + 2- z 4 a -2 -2 z 2 a -2$ (db)
Kauffman polynomial	$2 a z 9 + 2 z 9 a -1 + 6 a 2 z 8 + 3 z 8 a -2 + 9 z 8 + 6 a 3 z 7 + 2 a z 7 -3 z 7 a -1 + z 7 a -3 + 3 a 4 z 6 -20 a 2 z 6 -13 z 6 a -2 -36 z 6 + a 5 z 5 -16 a 3 z 5 -25 a z 5 -12 z 5 a -1 -4 z 5 a -3 - a 4 z 4 + 27 a 2 z 4 + 17 z 4 a -2 + 45 z 4 + 3 a 5 z 3 + 17 a 3 z 3 + 30 a z 3 + 21 z 3 a -1 + 5 z 3 a -3 + 3 a 6 z 2 -3 a 4 z 2 -26 a 2 z 2 -8 z 2 a -2 -28 z 2 - a 5 z -10 a 3 z -14 a z -7 z a -1 -2 z a -3 - a 6 + 4 a 4 + 12 a 2 + 2 a -2 + 10 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 L11n372. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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