L11n271

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 u 3 + 2 v u 2 -4 v w u 2 + 2 w u 2 -5 u 2 -2 v u + 5 v w u -2 w u + 4 u -2 v w$ (db)
Jones polynomial	$- q 10 + 2 q 9 -5 q 8 + 7 q 7 -9 q 6 + 11 q 5 -8 q 4 + 9 q 3 -5 q 2 + 3 q$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-2 z 4 a -4 -2 z 4 a -6 + 3 z 2 a -2 -3 z 2 a -4 -4 z 2 a -6 + 3 z 2 a -8 + 3 a -2 -2 a -4 -5 a -6 + 5 a -8 - a -10 + a -2 z -2 - a -4 z -2 -2 a -6 z -2 + 3 a -8 z -2 - a -10 z -2$ (db)
Kauffman polynomial	$z 9 a -7 + z 9 a -9 + 5 z 8 a -6 + 7 z 8 a -8 + 2 z 8 a -10 + 8 z 7 a -5 + 10 z 7 a -7 + 3 z 7 a -9 + z 7 a -11 + 7 z 6 a -4 -8 z 6 a -6 -23 z 6 a -8 -8 z 6 a -10 + 3 z 5 a -3 -19 z 5 a -5 -43 z 5 a -7 -26 z 5 a -9 -5 z 5 a -11 -12 z 4 a -4 - z 4 a -6 + 20 z 4 a -8 + 9 z 4 a -10 + 18 z 3 a -5 + 52 z 3 a -7 + 43 z 3 a -9 + 9 z 3 a -11 + 6 z 2 a -2 + 9 z 2 a -4 -3 z 2 a -6 -8 z 2 a -8 -2 z 2 a -10 + 4 z a -3 -10 z a -5 -34 z a -7 -27 z a -9 -7 z a -11 -4 a -2 -3 a -4 + 4 a -6 + 5 a -8 + a -10 -2 a -3 z -1 + 2 a -5 z -1 + 10 a -7 z -1 + 8 a -9 z -1 + 2 a -11 z -1 + a -2 z -2 + a -4 z -2 -2 a -6 z -2 -3 a -8 z -2 - a -10 z -2$ (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 L11n271. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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