L10a57

(Knotscape image)

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

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Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_14,6,15,5 X_16,11,17,12 X_20,17,7,18 X_18,14,19,13 X_12,20,13,19 X_4,16,5,15 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -9, 2, -8, 3, -10}, {9, -1, 10, -2, 4, -7, 6, -3, 8, -4, 5, -6, 7, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 4 + u 4 - v 2 u 3 + 5 v u 3 -3 u 3 + 3 v 2 u 2 -9 v u 2 + 3 u 2 -3 v 2 u + 5 v u - u + v 2 - v$ (db)
Jones polynomial	$-q^{7/2}+3 q^{5/2}-6 q^{3/2}+9 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 5 + z a 5 + a 5 z -1 - z 5 a 3 -2 z 3 a 3 -4 z a 3 -2 a 3 z -1 - z 5 a + 2 z a + 2 a z -1 + 2 z 3 a -1 + z a -1 - a -1 z -1 - z a -3$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 -4 a 4 z 8 -7 a 2 z 8 -3 z 8 -5 a 5 z 7 -9 a 3 z 7 -8 a z 7 -4 z 7 a -1 -3 a 6 z 6 + 6 a 4 z 6 + 12 a 2 z 6 -3 z 6 a -2 - a 7 z 5 + 12 a 5 z 5 + 27 a 3 z 5 + 21 a z 5 + 6 z 5 a -1 - z 5 a -3 + 6 a 6 z 4 -3 a 4 z 4 -9 a 2 z 4 + 6 z 4 a -2 + 6 z 4 + 2 a 7 z 3 -11 a 5 z 3 -31 a 3 z 3 -22 a z 3 -2 z 3 a -1 + 2 z 3 a -3 -2 a 6 z 2 + 3 a 2 z 2 -3 z 2 a -2 -2 z 2 + 5 a 5 z + 14 a 3 z + 12 a z + 2 z a -1 - z a -3 - a 2 - a 5 z -1 -2 a 3 z -1 -2 a z -1 - a -1 z -1$ (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 L10a57. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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