L11a461

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,4,15,3 X_18,15,19,16 X_16,6,17,5 X_12,18,5,17 X_8,22,9,21 X_20,8,21,7 X_22,10,13,9 X_10,14,11,13 X_2,11,3,12 X_4,20,1,19
Gauss code	{1, -10, 2, -11}, {4, -1, 7, -6, 8, -9, 10, -5}, {9, -2, 3, -4, 5, -3, 11, -7, 6, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 3 u 3 -2 v 2 u 3 - v 3 w u 3 + 2 v 2 w u 3 - v w u 3 -2 v 3 u 2 + 4 v 2 u 2 -2 v u 2 + 2 v 3 w u 2 -4 v 2 w u 2 + 3 v w u 2 - w u 2 + v 3 u -3 v 2 u + 4 v u + 2 v 2 w u -4 v w u + 2 w u -2 u + v 2 -2 v + 2 v w - w + 1$ (db)
Jones polynomial	$q 10 -3 q 9 + 7 q 8 -10 q 7 + 14 q 6 -15 q 5 + 16 q 4 -13 q 3 + 10 q 2 -6 q + 4- q -1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z 8 a -4 - z 6 a -2 + 5 z 6 a -4 -2 z 6 a -6 -3 z 4 a -2 + 8 z 4 a -4 -8 z 4 a -6 + z 4 a -8 + 5 z 2 a -4 -9 z 2 a -6 + 3 z 2 a -8 + 2 a -2 + a -4 -5 a -6 + 2 a -8 + a -4 z -2 -2 a -6 z -2 + a -8 z -2$ (db)
Kauffman polynomial	$2 z 10 a -4 + 2 z 10 a -6 + 5 z 9 a -3 + 10 z 9 a -5 + 5 z 9 a -7 + 4 z 8 a -2 + 2 z 8 a -4 + 5 z 8 a -6 + 7 z 8 a -8 + z 7 a -1 -19 z 7 a -3 -33 z 7 a -5 -6 z 7 a -7 + 7 z 7 a -9 -16 z 6 a -2 -28 z 6 a -4 -29 z 6 a -6 -11 z 6 a -8 + 6 z 6 a -10 -3 z 5 a -1 + 19 z 5 a -3 + 29 z 5 a -5 -4 z 5 a -7 -8 z 5 a -9 + 3 z 5 a -11 + 16 z 4 a -2 + 36 z 4 a -4 + 37 z 4 a -6 + 8 z 4 a -8 -8 z 4 a -10 + z 4 a -12 + z 3 a -1 -5 z 3 a -3 - z 3 a -5 + 9 z 3 a -7 + 2 z 3 a -9 -2 z 3 a -11 -2 z 2 a -2 -13 z 2 a -4 -24 z 2 a -6 -6 z 2 a -8 + 6 z 2 a -10 - z 2 a -12 + z a -3 -8 z a -5 -8 z a -7 + z a -9 -2 a -2 + 3 a -4 + 9 a -6 + 3 a -8 -2 a -10 + 2 a -5 z -1 + 2 a -7 z -1 - a -4 z -2 -2 a -6 z -2 - a -8 z -2$ (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 =

4 is the signature of L11a461. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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