L11n140

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n140 at Knotilus!
	[edit L11n140 Quick Notes]

[edit L11n140 Further Notes and Views]

Link Presentations

[edit Notes on L11n140's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_18,11,19,12 X_3,10,4,11 X_17,3,18,2 X_12,5,13,6 X₆₇₁₈ X_9,16,10,17 X_15,20,16,21 X_13,22,14,7 X_21,14,22,15 X_4,20,5,19
Gauss code	{1, 4, -3, -11, 5, -6}, {6, -1, -7, 3, 2, -5, -9, 10, -8, 7, -4, -2, 11, 8, -10, 9}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{2}-u^{2}v-3uv^{2}+5uv-3u-v+1}{uv}}$ (db)
Jones polynomial	${\frac {5}{q^{9/2}}}-{\frac {5}{q^{7/2}}}+{\frac {3}{q^{5/2}}}-{\frac {2}{q^{3/2}}}+{\frac {2}{q^{17/2}}}-{\frac {3}{q^{15/2}}}+{\frac {4}{q^{13/2}}}-{\frac {6}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{9}z^{-1}-z^{3}a^{7}+2a^{7}z^{-1}+z^{5}a^{5}+3z^{3}a^{5}+3za^{5}-2z^{3}a^{3}-4za^{3}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$3a^{10}z^{4}-7a^{10}z^{2}+2a^{10}+a^{9}z^{7}-a^{9}z^{5}-2a^{9}z^{3}+2a^{9}z-a^{9}z^{-1}+a^{8}z^{8}-3a^{8}z^{6}+9a^{8}z^{4}-13a^{8}z^{2}+5a^{8}+3a^{7}z^{7}-7a^{7}z^{5}+6a^{7}z^{3}+2a^{7}z-2a^{7}z^{-1}+a^{6}z^{8}-2a^{6}z^{6}+6a^{6}z^{4}-6a^{6}z^{2}+3a^{6}+2a^{5}z^{7}-6a^{5}z^{5}+11a^{5}z^{3}-5a^{5}z+a^{4}z^{6}-a^{4}+3a^{3}z^{3}-5a^{3}z+a^{3}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

2

-4

2

1

-1

-6

3

1

2

-8

3

0

-10

3

2

1

-12

1

3

2

-14

2

3

-1

-16

1

-18

2

-2

Integral Khovanov Homology

(db, data source)

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