L11n335

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_11,16,12,17 X₈₄₉₃ X_2,18,3,17 X_5,14,6,15 X_18,7,19,8 X_15,12,16,5 X_20,14,21,13 X_22,9,13,10 X_10,21,11,22 X_4,19,1,20
Gauss code	{1, -4, 3, -11}, {-5, -1, 6, -3, 9, -10, -2, 7}, {8, 5, -7, 2, 4, -6, 11, -8, 10, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 2 -3 v u 2 - v 2 w u 2 + 3 v w u 2 -2 w u 2 + 2 u 2 -3 v 2 u + 6 v u + 3 v 2 w u -6 v w u + 3 w u -3 u + 2 v 2 -3 v -2 v 2 w + 3 v w - w + 1$ (db)
Jones polynomial	$- q 2 + 5 q -9 + 13 q -1 -16 q -2 + 17 q -3 -14 q -4 + 12 q -5 -6 q -6 + 3 q -7$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a 8 z -2 + 2 z 2 a 6 -2 a 6 z -2 - a 6 -3 z 4 a 4 -4 z 2 a 4 + a 4 z -2 + z 6 a 2 + 2 z 4 a 2 + 2 z 2 a 2 - z 4 + 1$ (db)
Kauffman polynomial	$2 a 5 z 9 + 2 a 3 z 9 + 5 a 6 z 8 + 12 a 4 z 8 + 7 a 2 z 8 + 3 a 7 z 7 + 9 a 5 z 7 + 15 a 3 z 7 + 9 a z 7 -9 a 6 z 6 -18 a 4 z 6 -4 a 2 z 6 + 5 z 6 -3 a 7 z 5 -23 a 5 z 5 -36 a 3 z 5 -15 a z 5 + z 5 a -1 + 6 a 8 z 4 + 16 a 6 z 4 + 7 a 4 z 4 -9 a 2 z 4 -6 z 4 + 7 a 7 z 3 + 21 a 5 z 3 + 19 a 3 z 3 + 5 a z 3 -11 a 8 z 2 -16 a 6 z 2 -3 a 4 z 2 + 2 a 2 z 2 -9 a 7 z -9 a 5 z + 6 a 8 + 9 a 6 + 3 a 4 + 1 + 2 a 7 z -1 + 2 a 5 z -1 - a 8 z -2 -2 a 6 z -2 - a 4 z -2$ (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 =

-2 is the signature of L11n335. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$
$r = -6$	${\mathbb Z}^{3}$	${\mathbb Z}^{2}$
$r = -5$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 0$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{8}$
$r = 1$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$