L11a405

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 2 v u 4 - v w u 4 + w u 4 - u 4 + 2 v 2 u 3 -4 v u 3 - v 2 w u 3 + 4 v w u 3 -2 w u 3 + u 3 -2 v 2 u 2 + 4 v u 2 + v 2 w u 2 -4 v w u 2 + 2 w u 2 - u 2 + 2 v 2 u -4 v u - v 2 w u + 4 v w u -2 w u + u - v 2 + v + v 2 w -2 v w + w$ (db)
Jones polynomial	$1-3 q -1 + 7 q -2 -10 q -3 + 16 q -4 -16 q -5 + 18 q -6 -15 q -7 + 11 q -8 -7 q -9 + 3 q -10 - q -11$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- z 2 a 10 - a 10 z -2 -2 a 10 + 3 z 4 a 8 + 9 z 2 a 8 + 4 a 8 z -2 + 9 a 8 -2 z 6 a 6 -8 z 4 a 6 -14 z 2 a 6 -5 a 6 z -2 -14 a 6 - z 6 a 4 - z 4 a 4 + 5 z 2 a 4 + 2 a 4 z -2 + 7 a 4 + z 4 a 2 + 2 z 2 a 2$ (db)
Kauffman polynomial	$z 5 a 13 -2 z 3 a 13 + z a 13 + 3 z 6 a 12 -5 z 4 a 12 + z 2 a 12 + 5 z 7 a 11 -8 z 5 a 11 + 4 z 3 a 11 -4 z a 11 + a 11 z -1 + 6 z 8 a 10 -11 z 6 a 10 + 11 z 4 a 10 -10 z 2 a 10 - a 10 z -2 + 5 a 10 + 4 z 9 a 9 -18 z 5 a 9 + 33 z 3 a 9 -21 z a 9 + 5 a 9 z -1 + z 10 a 8 + 13 z 8 a 8 -48 z 6 a 8 + 72 z 4 a 8 -48 z 2 a 8 -4 a 8 z -2 + 18 a 8 + 8 z 9 a 7 -14 z 7 a 7 -6 z 5 a 7 + 36 z 3 a 7 -29 z a 7 + 9 a 7 z -1 + z 10 a 6 + 12 z 8 a 6 -50 z 6 a 6 + 77 z 4 a 6 -57 z 2 a 6 -5 a 6 z -2 + 21 a 6 + 4 z 9 a 5 -6 z 7 a 5 -5 z 5 a 5 + 13 z 3 a 5 -13 z a 5 + 5 a 5 z -1 + 5 z 8 a 4 -15 z 6 a 4 + 18 z 4 a 4 -18 z 2 a 4 -2 a 4 z -2 + 9 a 4 + 3 z 7 a 3 -8 z 5 a 3 + 4 z 3 a 3 + z 6 a 2 -3 z 4 a 2 + 2 z 2 a 2$ (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 L11a405. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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