L11a393

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v u 5 - v w u 5 + w u 5 - u 5 -2 v u 4 + v w u 4 -2 w u 4 + u 4 + 2 v u 3 - v w u 3 + 2 w u 3 - u 3 -2 v u 2 + v w u 2 -2 w u 2 + u 2 + 2 v u - v w u + 2 w u - u - v + v w -2 w + 1$ (db)
Jones polynomial	$- q 9 + 3 q 8 -6 q 7 + 7 q 6 -10 q 5 + 11 q 4 -9 q 3 + 9 q 2 -5 q + 5- q -1 + q -2$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z 8 a -4 -2 z 6 a -2 + 6 z 6 a -4 - z 6 a -6 -11 z 4 a -2 + 14 z 4 a -4 -4 z 4 a -6 + z 4 -21 z 2 a -2 + 20 z 2 a -4 -5 z 2 a -6 + 5 z 2 -20 a -2 + 18 a -4 -5 a -6 + 7-8 a -2 z -2 + 7 a -4 z -2 -2 a -6 z -2 + 3 z -2$ (db)
Kauffman polynomial	$z 10 a -2 + z 10 a -4 + z 9 a -1 + 5 z 9 a -3 + 4 z 9 a -5 - z 8 a -2 + 5 z 8 a -4 + 7 z 8 a -6 + z 8 -3 z 7 a -1 -18 z 7 a -3 -8 z 7 a -5 + 7 z 7 a -7 -15 z 6 a -2 -34 z 6 a -4 -19 z 6 a -6 + 7 z 6 a -8 -7 z 6 -5 z 5 a -1 + 2 z 5 a -3 -9 z 5 a -5 -10 z 5 a -7 + 6 z 5 a -9 + 42 z 4 a -2 + 51 z 4 a -4 + 16 z 4 a -6 -8 z 4 a -8 + 3 z 4 a -10 + 18 z 4 + 23 z 3 a -1 + 44 z 3 a -3 + 25 z 3 a -5 -3 z 3 a -7 -6 z 3 a -9 + z 3 a -11 -47 z 2 a -2 -37 z 2 a -4 -12 z 2 a -6 -22 z 2 -24 z a -1 -45 z a -3 -21 z a -5 + 3 z a -7 + 3 z a -9 + 28 a -2 + 22 a -4 + 7 a -6 + a -8 + 13 + 8 a -1 z -1 + 15 a -3 z -1 + 7 a -5 z -1 - a -7 z -1 - a -9 z -1 -8 a -2 z -2 -7 a -4 z -2 -2 a -6 z -2 -3 z -2$ (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 L11a393. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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