L11n351

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 + 2 v u 3 + v 2 w u 3 - v w u 3 - u 3 + 3 v 2 u 2 -4 v u 2 -3 v 2 w u 2 + 4 v w u 2 + u 2 -4 v 2 u + 3 v u - v 3 w u + 4 v 2 w u -3 v w u + v 2 - v + v 3 w -2 v 2 w + v w$ (db)
Jones polynomial	$- q 5 + 3 q 4 -7 q 3 + 12 q 2 -13 q + 15-13 q -1 + 11 q -2 -6 q -3 + 3 q -4$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 6 + a 2 z 4 + 2 z 4 a -2 -2 z 4 -2 a 2 z 2 + 3 z 2 a -2 - z 2 a -4 - z 2 + 2 a 4 -5 a 2 + 2 a -2 - a -4 + 2 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$2 a z 9 + 2 z 9 a -1 + 5 a 2 z 8 + 5 z 8 a -2 + 10 z 8 + 3 a 3 z 7 + 5 a z 7 + 7 z 7 a -1 + 5 z 7 a -3 -12 a 2 z 6 -5 z 6 a -2 + 3 z 6 a -4 -20 z 6 -3 a 3 z 5 -13 a z 5 -18 z 5 a -1 -7 z 5 a -3 + z 5 a -5 + 6 a 4 z 4 + 25 a 2 z 4 -5 z 4 a -4 + 24 z 4 + 4 a 3 z 3 + 12 a z 3 + 12 z 3 a -1 + 2 z 3 a -3 -2 z 3 a -5 -11 a 4 z 2 -29 a 2 z 2 - z 2 a -2 + 3 z 2 a -4 -22 z 2 -5 a 3 z -8 a z -3 z a -1 + z a -3 + z a -5 + 6 a 4 + 13 a 2 - a -4 + 9 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 L11n351. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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