L11n310

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 4 - u 4 -3 v u 3 - v 2 w u 3 + 3 v w u 3 - w u 3 + 2 u 3 - v 2 u 2 + 4 v u 2 + 2 v 2 w u 2 -4 v w u 2 + w u 2 -2 u 2 + v 2 u -3 v u -2 v 2 w u + 3 v w u + u + v 2 w - v w$ (db)
Jones polynomial	$- q 8 + 3 q 7 -7 q 6 + 10 q 5 -12 q 4 + 14 q 3 -11 q 2 + 10 q -5 + 3 q -1$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -4 -4 z 4 a -2 + 4 z 4 a -4 - z 4 a -6 -12 z 2 a -2 + 10 z 2 a -4 -2 z 2 a -6 + 3 z 2 -13 a -2 + 11 a -4 -3 a -6 + 5-5 a -2 z -2 + 4 a -4 z -2 - a -6 z -2 + 2 z -2$ (db)
Kauffman polynomial	$2 z 9 a -3 + 2 z 9 a -5 + 5 z 8 a -2 + 10 z 8 a -4 + 5 z 8 a -6 + 3 z 7 a -1 + 3 z 7 a -3 + 5 z 7 a -5 + 5 z 7 a -7 -17 z 6 a -2 -29 z 6 a -4 -9 z 6 a -6 + 3 z 6 a -8 -6 z 5 a -1 -18 z 5 a -3 -22 z 5 a -5 -9 z 5 a -7 + z 5 a -9 + 37 z 4 a -2 + 44 z 4 a -4 + 8 z 4 a -6 -5 z 4 a -8 + 6 z 4 + 10 z 3 a -1 + 35 z 3 a -3 + 32 z 3 a -5 + 5 z 3 a -7 -2 z 3 a -9 -41 z 2 a -2 -36 z 2 a -4 -8 z 2 a -6 + z 2 a -8 -14 z 2 -13 z a -1 -29 z a -3 -21 z a -5 -4 z a -7 + z a -9 + 21 a -2 + 18 a -4 + 5 a -6 + 9 + 5 a -1 z -1 + 9 a -3 z -1 + 5 a -5 z -1 + a -7 z -1 -5 a -2 z -2 -4 a -4 z -2 - a -6 z -2 -2 z -2$ (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 L11n310. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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