L11a421

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 4 - w u 4 + u 4 -2 v 2 u 3 + 5 v u 3 -3 v w u 3 + 4 w u 3 -3 u 3 + 5 v 2 u 2 -7 v u 2 -3 v 2 w u 2 + 7 v w u 2 -5 w u 2 + 3 u 2 -4 v 2 u + 3 v u + 3 v 2 w u -5 v w u + 2 w u + v 2 - v 2 w + v w$ (db)
Jones polynomial	$q -2 -4 q -3 + 10 q -4 -15 q -5 + 21 q -6 -22 q -7 + 23 q -8 -18 q -9 + 14 q -10 -8 q -11 + 3 q -12 - q -13$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- a 14 z -2 + 4 a 12 z -2 + 5 a 12 -9 z 2 a 10 -5 a 10 z -2 -13 a 10 + 5 z 4 a 8 + 9 z 2 a 8 + 2 a 8 z -2 + 7 a 8 + 4 z 4 a 6 + 5 z 2 a 6 + a 6 + z 4 a 4$ (db)
Kauffman polynomial	$z 7 a 15 -4 z 5 a 15 + 6 z 3 a 15 -4 z a 15 + a 15 z -1 + 3 z 8 a 14 -10 z 6 a 14 + 12 z 4 a 14 -7 z 2 a 14 - a 14 z -2 + 3 a 14 + 4 z 9 a 13 -7 z 7 a 13 -9 z 5 a 13 + 26 z 3 a 13 -19 z a 13 + 5 a 13 z -1 + 2 z 10 a 12 + 9 z 8 a 12 -44 z 6 a 12 + 52 z 4 a 12 -29 z 2 a 12 -4 a 12 z -2 + 15 a 12 + 13 z 9 a 11 -23 z 7 a 11 -17 z 5 a 11 + 48 z 3 a 11 -33 z a 11 + 9 a 11 z -1 + 2 z 10 a 10 + 22 z 8 a 10 -71 z 6 a 10 + 69 z 4 a 10 -42 z 2 a 10 -5 a 10 z -2 + 20 a 10 + 9 z 9 a 9 -35 z 5 a 9 + 35 z 3 a 9 -18 z a 9 + 5 a 9 z -1 + 16 z 8 a 8 -27 z 6 a 8 + 19 z 4 a 8 -15 z 2 a 8 -2 a 8 z -2 + 8 a 8 + 15 z 7 a 7 -19 z 5 a 7 + 7 z 3 a 7 + 10 z 6 a 6 -9 z 4 a 6 + 5 z 2 a 6 - a 6 + 4 z 5 a 5 + z 4 a 4$ (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 =

-4 is the signature of L11a421. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -5$	$i = -3$
$r = -11$	${\mathbb Z}$
$r = -10$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -9$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -8$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -7$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -6$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{12}$
$r = -5$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = -4$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{12}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = -2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$