L11n257

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,16,12,17 X_21,18,22,19 X_13,20,14,21 X_19,12,20,13 X_17,22,18,9 X_8,16,5,15 X_14,8,15,7 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 9, -8}, {11, -2, -3, 6, -5, -9, 8, 3, -7, 4, -6, 5, -4, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 4 v u 2 - v w u 2 + 3 w u 2 - u 2 -3 v u + v w u -4 w u + u + 2 w$ (db)
Jones polynomial	$2 q -1 -2 q -2 + 6 q -3 -6 q -4 + 8 q -5 -6 q -6 + 6 q -7 -5 q -8 + 2 q -9 - q -10$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 10 z -2 - a 10 + 3 z 2 a 8 + 3 a 8 z -2 + 5 a 8 -2 z 4 a 6 -5 z 2 a 6 -2 a 6 z -2 -5 a 6 - z 4 a 4 - z 2 a 4 - a 4 z -2 -2 a 4 + 2 z 2 a 2 + a 2 z -2 + 3 a 2$ (db)
Kauffman polynomial	$z 7 a 11 -5 z 5 a 11 + 9 z 3 a 11 -7 z a 11 + 2 a 11 z -1 + 2 z 8 a 10 -8 z 6 a 10 + 8 z 4 a 10 -2 z 2 a 10 - a 10 z -2 + a 10 + z 9 a 9 + 2 z 7 a 9 -24 z 5 a 9 + 40 z 3 a 9 -27 z a 9 + 8 a 9 z -1 + 6 z 8 a 8 -23 z 6 a 8 + 22 z 4 a 8 -7 z 2 a 8 -3 a 8 z -2 + 5 a 8 + z 9 a 7 + 5 z 7 a 7 -33 z 5 a 7 + 49 z 3 a 7 -34 z a 7 + 10 a 7 z -1 + 4 z 8 a 6 -13 z 6 a 6 + 11 z 4 a 6 -4 z 2 a 6 -2 a 6 z -2 + 4 a 6 + 4 z 7 a 5 -13 z 5 a 5 + 17 z 3 a 5 -10 z a 5 + 2 a 5 z -1 + 2 z 6 a 4 -3 z 4 a 4 + 4 z 2 a 4 + a 4 z -2 -3 a 4 + z 5 a 3 - z 3 a 3 + 4 z a 3 -2 a 3 z -1 + 3 z 2 a 2 + a 2 z -2 -4 a 2$ (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 =

-2 is the signature of L11n257. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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