L11a436

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + 3 v 2 u 3 -2 v u 3 -2 v 2 w u 3 + 3 v w u 3 - w u 3 + v 3 u 2 -4 v 2 u 2 + 3 v u 2 + 3 v 2 w u 2 -4 v w u 2 + w u 2 - v 3 u + 4 v 2 u -3 v u -3 v 2 w u + 4 v w u - w u + v 3 -3 v 2 + 2 v + 2 v 2 w -3 v w + w$ (db)
Jones polynomial	$1-2 q -1 + 6 q -2 -10 q -3 + 16 q -4 -17 q -5 + 19 q -6 -16 q -7 + 13 q -8 -8 q -9 + 3 q -10 - q -11$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- z 2 a 10 - a 10 z -2 -2 a 10 + 3 z 4 a 8 + 8 z 2 a 8 + 4 a 8 z -2 + 8 a 8 -2 z 6 a 6 -7 z 4 a 6 -10 z 2 a 6 -5 a 6 z -2 -10 a 6 - z 6 a 4 -2 z 4 a 4 + 2 a 4 z -2 + 2 a 4 + z 4 a 2 + 3 z 2 a 2 + 2 a 2$ (db)
Kauffman polynomial	$z 5 a 13 -2 z 3 a 13 + z a 13 + 3 z 6 a 12 -4 z 4 a 12 + z 2 a 12 + 6 z 7 a 11 -10 z 5 a 11 + 8 z 3 a 11 -5 z a 11 + a 11 z -1 + 7 z 8 a 10 -11 z 6 a 10 + 10 z 4 a 10 -11 z 2 a 10 - a 10 z -2 + 6 a 10 + 4 z 9 a 9 + 5 z 7 a 9 -29 z 5 a 9 + 41 z 3 a 9 -25 z a 9 + 5 a 9 z -1 + z 10 a 8 + 13 z 8 a 8 -39 z 6 a 8 + 55 z 4 a 8 -45 z 2 a 8 -4 a 8 z -2 + 21 a 8 + 7 z 9 a 7 -6 z 7 a 7 -18 z 5 a 7 + 43 z 3 a 7 -35 z a 7 + 9 a 7 z -1 + z 10 a 6 + 9 z 8 a 6 -32 z 6 a 6 + 49 z 4 a 6 -44 z 2 a 6 -5 a 6 z -2 + 22 a 6 + 3 z 9 a 5 -3 z 7 a 5 -5 z 5 a 5 + 14 z 3 a 5 -15 z a 5 + 5 a 5 z -1 + 3 z 8 a 4 -6 z 6 a 4 + 4 z 4 a 4 -6 z 2 a 4 -2 a 4 z -2 + 6 a 4 + 2 z 7 a 3 -5 z 5 a 3 + 2 z 3 a 3 + z a 3 + z 6 a 2 -4 z 4 a 2 + 5 z 2 a 2 -2 a 2$ (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 =

-4 is the signature of L11a436. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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