L11n457

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_15,18,16,11 X_20,9,21,10 X_22,13,19,14 X_14,21,15,22 X_10,19,5,20 X_17,8,18,9 X_7,16,8,17 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {7, -4, 6, -5}, {10, -1, -9, 8, 4, -7}, {11, -2, 5, -6, -3, 9, -8, 3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 2 - v u 2 - v 2 w u 2 + v w u 2 - v 2 x u 2 + v x u 2 - v w x u 2 - v 2 u + 3 v u + v 2 w u -2 v w u + w u + v 2 x u -2 v x u - v 2 w x u + 3 v w x u - w x u + x u - u - v + v w - w + v x - v w x + w x - x$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{12}{q^{13/2}}+\frac{10}{q^{15/2}}-\frac{11}{q^{17/2}}+\frac{6}{q^{19/2}}-\frac{5}{q^{21/2}}+\frac{1}{q^{23/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a 13 z -3 -2 z a 11 -5 a 11 z -1 -3 a 11 z -3 + 5 z 3 a 9 + 14 z a 9 + 10 a 9 z -1 + 3 a 9 z -3 -3 z 5 a 7 -11 z 3 a 7 -12 z a 7 -5 a 7 z -1 - a 7 z -3 - z 5 a 5 -2 z 3 a 5$ (db)
Kauffman polynomial	$- z 4 a 14 -5 z 5 a 13 + 5 z 3 a 13 + 4 z a 13 -5 a 13 z -1 + a 13 z -3 - z 8 a 12 -2 z 6 a 12 + 3 z 4 a 12 -6 z 2 a 12 -3 a 12 z -2 + 10 a 12 - z 9 a 11 - z 7 a 11 - z 5 a 11 -3 z 3 a 11 + 15 z a 11 -12 a 11 z -1 + 3 a 11 z -3 -5 z 8 a 10 + 6 z 6 a 10 + 4 z 4 a 10 -20 z 2 a 10 -6 a 10 z -2 + 19 a 10 - z 9 a 9 -7 z 7 a 9 + 22 z 5 a 9 -29 z 3 a 9 + 23 z a 9 -12 a 9 z -1 + 3 a 9 z -3 -4 z 8 a 8 + 5 z 6 a 8 + 5 z 4 a 8 -14 z 2 a 8 -3 a 8 z -2 + 10 a 8 -6 z 7 a 7 + 17 z 5 a 7 -19 z 3 a 7 + 12 z a 7 -5 a 7 z -1 + a 7 z -3 -3 z 6 a 6 + 5 z 4 a 6 - z 5 a 5 + 2 z 3 a 5$ (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 =

-5 is the signature of L11n457. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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