L11n307

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$u 4 + 2 v u 3 - v w u 3 + w u 3 -3 u 3 + v 2 u 2 -3 v u 2 -3 v 2 w u 2 + 3 v w u 2 - w u 2 + 3 u 2 - v 2 u + v u + 3 v 2 w u -2 v w u - v 2 w$ (db)
Jones polynomial	$- q 10 + 3 q 9 -6 q 8 + 8 q 7 -9 q 6 + 11 q 5 -8 q 4 + 8 q 3 -4 q 2 + 2 q$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$- z 4 a -4 -3 z 4 a -6 + 2 z 2 a -2 + 2 z 2 a -4 -9 z 2 a -6 + 4 z 2 a -8 + a -2 + 5 a -4 -12 a -6 + 7 a -8 - a -10 + 2 a -4 z -2 -5 a -6 z -2 + 4 a -8 z -2 - a -10 z -2$ (db)
Kauffman polynomial	$2 z 9 a -7 + 2 z 9 a -9 + 7 z 8 a -6 + 10 z 8 a -8 + 3 z 8 a -10 + 7 z 7 a -5 + 5 z 7 a -7 - z 7 a -9 + z 7 a -11 + 3 z 6 a -4 -24 z 6 a -6 -39 z 6 a -8 -12 z 6 a -10 + z 5 a -3 -20 z 5 a -5 -38 z 5 a -7 -21 z 5 a -9 -4 z 5 a -11 + z 4 a -4 + 35 z 4 a -6 + 48 z 4 a -8 + 14 z 4 a -10 + 3 z 3 a -3 + 25 z 3 a -5 + 52 z 3 a -7 + 36 z 3 a -9 + 6 z 3 a -11 + 3 z 2 a -2 -8 z 2 a -4 -37 z 2 a -6 -32 z 2 a -8 -6 z 2 a -10 -18 z a -5 -33 z a -7 -19 z a -9 -4 z a -11 - a -2 + 8 a -4 + 20 a -6 + 15 a -8 + 3 a -10 + 5 a -5 z -1 + 9 a -7 z -1 + 5 a -9 z -1 + a -11 z -1 -2 a -4 z -2 -5 a -6 z -2 -4 a -8 z -2 - a -10 z -2$ (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 L11n307. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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