L11n312

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_3,13,4,12 X_9,20,10,21 X_7,16,8,17 X_13,18,14,19 X_19,14,20,15 X_15,22,16,11 X_17,10,18,5 X_21,8,22,9 X₂₅₃₆ X_11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -4, 9, -3, 8}, {-11, 2, -5, 6, -7, 4, -8, 5, -6, 3, -9, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 4 - v w u 4 - u 4 + v 2 u 3 -3 v u 3 - v 2 w u 3 + 2 v w u 3 + 2 u 3 + 3 v u 2 + 2 v 2 w u 2 -3 v w u 2 -2 u 2 -2 v u -2 v 2 w u + 3 v w u - w u + u + v + v 2 w - v w$ (db)
Jones polynomial	$1-3 q -1 + 7 q -2 -8 q -3 + 12 q -4 -11 q -5 + 11 q -6 -8 q -7 + 5 q -8 -2 q -9$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- a 10 z -2 - a 10 + z 4 a 8 + 4 z 2 a 8 + 4 a 8 z -2 + 7 a 8 - z 6 a 6 -4 z 4 a 6 -9 z 2 a 6 -5 a 6 z -2 -12 a 6 - z 6 a 4 -2 z 4 a 4 + 2 z 2 a 4 + 2 a 4 z -2 + 5 a 4 + z 4 a 2 + 2 z 2 a 2 + a 2$ (db)
Kauffman polynomial	$3 z 3 a 11 -4 z a 11 + a 11 z -1 + z 6 a 10 + 4 z 4 a 10 -6 z 2 a 10 - a 10 z -2 + 3 a 10 + 5 z 7 a 9 -15 z 5 a 9 + 28 z 3 a 9 -19 z a 9 + 5 a 9 z -1 + 6 z 8 a 8 -21 z 6 a 8 + 39 z 4 a 8 -32 z 2 a 8 -4 a 8 z -2 + 15 a 8 + 2 z 9 a 7 + 4 z 7 a 7 -27 z 5 a 7 + 46 z 3 a 7 -33 z a 7 + 9 a 7 z -1 + 10 z 8 a 6 -33 z 6 a 6 + 44 z 4 a 6 -37 z 2 a 6 -5 a 6 z -2 + 20 a 6 + 2 z 9 a 5 + 2 z 7 a 5 -20 z 5 a 5 + 25 z 3 a 5 -18 z a 5 + 5 a 5 z -1 + 4 z 8 a 4 -10 z 6 a 4 + 6 z 4 a 4 -8 z 2 a 4 -2 a 4 z -2 + 8 a 4 + 3 z 7 a 3 -8 z 5 a 3 + 4 z 3 a 3 + z 6 a 2 -3 z 4 a 2 + 3 z 2 a 2 - a 2$ (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 =

-4 is the signature of L11n312. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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