L11a420

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_20,11,21,12 X_22,15,11,16 X_14,21,15,22 X_8,18,9,17 X_16,8,17,7 X_18,10,19,9 X_10,20,5,19 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 7, -6, 8, -9}, {3, -2, 11, -5, 4, -7, 6, -8, 9, -3, 5, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 4 - v 2 w u 4 + v w u 4 -3 v 2 u 3 + 2 v u 3 + 2 v 2 w u 3 -3 v w u 3 + w u 3 + 3 v 2 u 2 -4 v u 2 -2 v 2 w u 2 + 4 v w u 2 -3 w u 2 + 2 u 2 - v 2 u + 3 v u -2 v w u + 3 w u -2 u - v - w + 1$ (db)
Jones polynomial	$q 7 -3 q 6 + 7 q 5 -10 q 4 + 14 q 3 -14 q 2 + 15 q -11 + 9 q -1 -5 q -2 + 2 q -3 - q -4$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$- z 8 a -2 -6 z 6 a -2 + z 6 a -4 + 2 z 6 - a 2 z 4 -15 z 4 a -2 + 4 z 4 a -4 + 10 z 4 -4 a 2 z 2 -21 z 2 a -2 + 6 z 2 a -4 + 18 z 2 -4 a 2 -16 a -2 + 5 a -4 + 15- a 2 z -2 -5 a -2 z -2 + 2 a -4 z -2 + 4 z -2$ (db)
Kauffman polynomial	$z 10 a -2 + z 10 + 2 a z 9 + 6 z 9 a -1 + 4 z 9 a -3 + 2 a 2 z 8 + 9 z 8 a -2 + 7 z 8 a -4 + 4 z 8 + a 3 z 7 -3 a z 7 -12 z 7 a -1 - z 7 a -3 + 7 z 7 a -5 -8 a 2 z 6 -35 z 6 a -2 -12 z 6 a -4 + 6 z 6 a -6 -25 z 6 -5 a 3 z 5 -12 a z 5 -11 z 5 a -1 -15 z 5 a -3 -8 z 5 a -5 + 3 z 5 a -7 + 10 a 2 z 4 + 47 z 4 a -2 + 11 z 4 a -4 -8 z 4 a -6 + z 4 a -8 + 37 z 4 + 8 a 3 z 3 + 28 a z 3 + 39 z 3 a -1 + 23 z 3 a -3 + 2 z 3 a -5 -2 z 3 a -7 -7 a 2 z 2 -39 z 2 a -2 -9 z 2 a -4 + 6 z 2 a -6 - z 2 a -8 -30 z 2 -5 a 3 z -21 a z -33 z a -1 -16 z a -3 + z a -5 + 4 a 2 + 20 a -2 + 6 a -4 -2 a -6 + 17 + a 3 z -1 + 5 a z -1 + 9 a -1 z -1 + 5 a -3 z -1 - a 2 z -2 -5 a -2 z -2 -2 a -4 z -2 -4 z -2$ (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 L11a420. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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