L11n301

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 4 + v 2 u 3 -3 v u 3 -2 v 2 w u 3 + 2 v w u 3 + u 3 - v 2 u 2 + 4 v u 2 + 2 v 2 w u 2 -4 v w u 2 + w u 2 -2 u 2 -2 v u - v 2 w u + 3 v w u - w u + 2 u - v w$ (db)
Jones polynomial	$- q 11 + 3 q 10 -6 q 9 + 9 q 8 -11 q 7 + 12 q 6 -10 q 5 + 9 q 4 -4 q 3 + 3 q 2$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-2 z 6 a -6 + 3 z 4 a -4 -10 z 4 a -6 + 3 z 4 a -8 + 10 z 2 a -4 -20 z 2 a -6 + 10 z 2 a -8 - z 2 a -10 + 9 a -4 -18 a -6 + 11 a -8 -2 a -10 + 2 a -4 z -2 -5 a -6 z -2 + 4 a -8 z -2 - a -10 z -2$ (db)
Kauffman polynomial	$z 9 a -7 + z 9 a -9 + 4 z 8 a -6 + 7 z 8 a -8 + 3 z 8 a -10 + 3 z 7 a -5 + 7 z 7 a -7 + 8 z 7 a -9 + 4 z 7 a -11 -14 z 6 a -6 -17 z 6 a -8 + 3 z 6 a -12 -9 z 5 a -5 -28 z 5 a -7 -26 z 5 a -9 -6 z 5 a -11 + z 5 a -13 + 6 z 4 a -4 + 32 z 4 a -6 + 23 z 4 a -8 -9 z 4 a -10 -6 z 4 a -12 + 18 z 3 a -5 + 48 z 3 a -7 + 33 z 3 a -9 + z 3 a -11 -2 z 3 a -13 -16 z 2 a -4 -38 z 2 a -6 -19 z 2 a -8 + 6 z 2 a -10 + 3 z 2 a -12 -19 z a -5 -35 z a -7 -19 z a -9 -2 z a -11 + z a -13 + 11 a -4 + 22 a -6 + 13 a -8 - a -12 + 5 a -5 z -1 + 9 a -7 z -1 + 5 a -9 z -1 + a -11 z -1 -2 a -4 z -2 -5 a -6 z -2 -4 a -8 z -2 - a -10 z -2$ (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 =

4 is the signature of L11n301. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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