L11n83

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n83 at Knotilus!
	[edit L11n83 Quick Notes]

[edit L11n83 Further Notes and Views]

Link Presentations

[edit Notes on L11n83's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_20,7,21,8 X_4,21,1,22 X_5,14,6,15 X_10,4,11,3 X_11,16,12,17 X_15,12,16,13 X_13,22,14,5 X_18,9,19,10 X_2,18,3,17 X_8,19,9,20
Gauss code	{1, -10, 5, -3}, {-4, -1, 2, -11, 9, -5, -6, 7, -8, 4, -7, 6, 10, -9, 11, -2, 3, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2(t(1)-1)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {2}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {6}{q^{5/2}}}-{\frac {8}{q^{7/2}}}+{\frac {8}{q^{9/2}}}-{\frac {8}{q^{11/2}}}+{\frac {5}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {2}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{9}z^{-1}-a^{7}z^{3}+2a^{7}z^{-1}+a^{5}z^{5}+2a^{5}z^{3}+a^{5}z-a^{5}z^{-1}+a^{3}z^{5}+2a^{3}z^{3}+a^{3}z+a^{3}z^{-1}-az^{3}-2az-az^{-1}$ (db)
Kauffman polynomial	$3a^{10}z^{4}-8a^{10}z^{2}+3a^{10}+a^{9}z^{7}-a^{9}z^{5}-a^{9}z^{3}+a^{9}z-a^{9}z^{-1}+2a^{8}z^{8}-8a^{8}z^{6}+19a^{8}z^{4}-20a^{8}z^{2}+7a^{8}+a^{7}z^{9}-a^{7}z^{7}-a^{7}z^{5}+5a^{7}z^{3}+2a^{7}z-2a^{7}z^{-1}+4a^{6}z^{8}-13a^{6}z^{6}+22a^{6}z^{4}-14a^{6}z^{2}+4a^{6}+a^{5}z^{9}-2a^{5}z^{5}+3a^{5}z^{3}+a^{5}z-a^{5}z^{-1}+2a^{4}z^{8}-3a^{4}z^{6}+2a^{4}z^{4}-2a^{4}z^{2}+2a^{3}z^{7}-a^{3}z^{5}-6a^{3}z^{3}+3a^{3}z-a^{3}z^{-1}+2a^{2}z^{6}-4a^{2}z^{4}+a^{2}+az^{5}-3az^{3}+3az-az^{-1}$ (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

1

2

χ

2

1

0

1

-1

-2

4

1

3

-4

4

3

-1

-6

4

2

-8

4

0

-10

4

0

-12

1

4

3

-14

2

4

-2

-16

1

-18

2

-2

Integral Khovanov Homology

(db, data source)

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