L11n275

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_2,14,3,13 X_14,7,15,8 X_9,18,10,19 X_17,11,18,22 X_11,21,12,20 X_21,17,22,16 X_15,1,16,4 X_19,10,20,5
Gauss code	{1, -4, -3, 10}, {-2, -1, 5, 3, -6, 11}, {-8, 2, 4, -5, -10, 9, -7, 6, -11, 8, -9, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v w u 4 + v 2 u 3 - v u 3 + 3 v w u 3 -2 w u 3 -2 v 2 u 2 + 3 v u 2 -3 v w u 2 + 2 w u 2 + 2 v 2 u -3 v u + v w u - w u + v$ (db)
Jones polynomial	$-2 q 3 + 4 q 2 -6 q + 9-8 q -1 + 9 q -2 -6 q -3 + 5 q -4 -2 q -5 + q -6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- a 2 z 6 + a 4 z 4 -5 a 2 z 4 + 3 z 4 + 3 a 4 z 2 -12 a 2 z 2 -2 z 2 a -2 + 10 z 2 + 4 a 4 -13 a 2 -3 a -2 + 12 + 2 a 4 z -2 -5 a 2 z -2 - a -2 z -2 + 4 z -2$ (db)
Kauffman polynomial	$a 3 z 9 + a z 9 + 3 a 4 z 8 + 6 a 2 z 8 + 3 z 8 + 2 a 5 z 7 + 3 a 3 z 7 + 4 a z 7 + 3 z 7 a -1 + a 6 z 6 -11 a 4 z 6 -22 a 2 z 6 + z 6 a -2 -9 z 6 -6 a 5 z 5 -20 a 3 z 5 -23 a z 5 -9 z 5 a -1 -4 a 6 z 4 + 15 a 4 z 4 + 36 a 2 z 4 + 2 z 4 a -2 + 19 z 4 + 3 a 5 z 3 + 28 a 3 z 3 + 43 a z 3 + 21 z 3 a -1 + 3 z 3 a -3 + 4 a 6 z 2 -15 a 4 z 2 -36 a 2 z 2 -4 z 2 a -2 -21 z 2 -18 a 3 z -33 a z -19 z a -1 -4 z a -3 - a 6 + 8 a 4 + 20 a 2 + 3 a -2 + 15 + 5 a 3 z -1 + 9 a z -1 + 5 a -1 z -1 + a -3 z -1 -2 a 4 z -2 -5 a 2 z -2 - a -2 z -2 -4 z -2$ (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 =

0 is the signature of L11n275. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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