L11n373

(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_21,15,22,14 X_9,20,10,21 X_13,9,14,8 X_15,17,16,22 X_17,5,18,16 X_19,12,20,13 X₂₅₃₆ X_11,4,12,1
Gauss code	{1, -10, -2, 11}, {-8, 3, -9, 5, -4, 7}, {10, -1, -3, 6, -5, 2, -11, 9, -6, 4, -7, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 w u 4 + v w u 4 - v 2 u 3 + 2 v u 3 + 2 v 2 w u 3 -3 v w u 3 + w u 3 + 2 v 2 u 2 -4 v u 2 - v 2 w u 2 + 4 v w u 2 -2 w u 2 + u 2 - v 2 u + 3 v u -2 v w u + w u -2 u - v + 1$ (db)
Jones polynomial	$- q 5 + 3 q 4 -6 q 3 + 10 q 2 -11 q + 13-11 q -1 + 9 q -2 -5 q -3 + 3 q -4$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z 8 - a 2 z 6 - z 6 a -2 + 6 z 6 -5 a 2 z 4 -4 z 4 a -2 + 13 z 4 + a 4 z 2 -10 a 2 z 2 -5 z 2 a -2 + 12 z 2 + 3 a 4 -8 a 2 - a -2 + 6 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$2 a z 9 + 2 z 9 a -1 + 5 a 2 z 8 + 4 z 8 a -2 + 9 z 8 + 3 a 3 z 7 + a z 7 + 2 z 7 a -1 + 4 z 7 a -3 -18 a 2 z 6 -5 z 6 a -2 + 3 z 6 a -4 -26 z 6 -6 a 3 z 5 -10 a z 5 -10 z 5 a -1 -5 z 5 a -3 + z 5 a -5 + 6 a 4 z 4 + 37 a 2 z 4 -6 z 4 a -4 + 37 z 4 + 7 a 3 z 3 + 18 a z 3 + 11 z 3 a -1 -2 z 3 a -3 -2 z 3 a -5 -14 a 4 z 2 -36 a 2 z 2 + z 2 a -2 + 3 z 2 a -4 -24 z 2 -7 a 3 z -10 a z -3 z a -1 + z a -3 + z a -5 + 7 a 4 + 14 a 2 - a -2 - a -4 + 8 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 =

0 is the signature of L11n373. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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