L11n110

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 4 v u 4 - u 4 -7 v u 3 + 6 u 3 + 6 v u 2 -7 u 2 - v u + 4 u -1$ (db)
Jones polynomial	$q^{9/2}-3 q^{7/2}+6 q^{5/2}-10 q^{3/2}+12 \sqrt{q}-\frac{13}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{3}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 - a 3 z 5 + 5 a z 5 -2 z 5 a -1 -4 a 3 z 3 + 11 a z 3 -7 z 3 a -1 + z 3 a -3 + a 5 z -8 a 3 z + 12 a z -9 z a -1 + 2 z a -3 + 2 a 5 z -1 -5 a 3 z -1 + 6 a z -1 -4 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 -5 a 2 z 8 -3 z 8 a -2 -8 z 8 -7 a 3 z 7 -14 a z 7 -10 z 7 a -1 -3 z 7 a -3 -3 a 4 z 6 + 5 a 2 z 6 + 3 z 6 a -2 - z 6 a -4 + 12 z 6 + 17 a 3 z 5 + 44 a z 5 + 36 z 5 a -1 + 9 z 5 a -3 + 9 z 4 a -2 + 3 z 4 a -4 + 6 z 4 -6 a 5 z 3 -29 a 3 z 3 -51 a z 3 -37 z 3 a -1 -9 z 3 a -3 - a 4 z 2 -3 a 2 z 2 -11 z 2 a -2 -3 z 2 a -4 -10 z 2 + 7 a 5 z + 21 a 3 z + 29 a z + 19 z a -1 + 4 z a -3 + a 4 + a 2 + 3 a -2 + a -4 + 3-2 a 5 z -1 -5 a 3 z -1 -6 a z -1 -4 a -1 z -1 - a -3 z -1$ (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 =

-1 is the signature of L11n110. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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