L11n311

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_13,19,14,18 X_17,11,18,22 X_7,17,8,16 X_21,8,22,9 X_9,20,10,21 X_15,5,16,10 X_19,15,20,14 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -5, 6, -7, 8}, {11, -2, -3, 9, -8, 5, -4, 3, -9, 7, -6, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 + v 2 u 3 + v u 3 - v w u 3 - u 3 -2 v u 2 - v 2 w u 2 + 2 v w u 2 + u 2 + v u + v 2 w u - v w u - w u - v w + w$ (db)
Jones polynomial	$- q 5 + 2 q 4 -3 q 3 + 3 q 2 -3 q + 3 + q -2 + 2 q -3 - q -4 + q -5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z 6 -2 a 2 z 4 + 7 z 4 + a 4 z 2 -10 a 2 z 2 -3 z 2 a -2 - z 2 a -4 + 14 z 2 + 3 a 4 -12 a 2 -4 a -2 + 13 + 2 a 4 z -2 -5 a 2 z -2 - a -2 z -2 + 4 z -2$ (db)
Kauffman polynomial	$a 3 z 9 + a z 9 + a 4 z 8 + 3 a 2 z 8 + z 8 a -2 + 3 z 8 -6 a 3 z 7 -5 a z 7 + 3 z 7 a -1 + 2 z 7 a -3 -7 a 4 z 6 -23 a 2 z 6 -4 z 6 a -2 + 2 z 6 a -4 -22 z 6 + 7 a 3 z 5 -3 a z 5 -18 z 5 a -1 -7 z 5 a -3 + z 5 a -5 + 16 a 4 z 4 + 53 a 2 z 4 + 6 z 4 a -2 -6 z 4 a -4 + 49 z 4 + 6 a 3 z 3 + 30 a z 3 + 33 z 3 a -1 + 6 z 3 a -3 -3 z 3 a -5 -17 a 4 z 2 -49 a 2 z 2 -8 z 2 a -2 + 2 z 2 a -4 -42 z 2 -13 a 3 z -29 a z -21 z a -1 -4 z a -3 + z a -5 + 9 a 4 + 21 a 2 + 5 a -2 + 18 + 5 a 3 z -1 + 9 a z -1 + 5 a -1 z -1 + a -3 z -1 -2 a 4 z -2 -5 a 2 z -2 - a -2 z -2 -4 z -2$ (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 =

0 is the signature of L11n311. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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