L11n267

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n267 at Knotilus!
	[edit L11n267 Quick Notes]

[edit L11n267 Further Notes and Views]

Link Presentations

[edit Notes on L11n267's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_14,7,15,8 X_8,13,5,14 X_11,19,12,18 X_15,21,16,20 X_17,9,18,22 X_21,17,22,16 X_19,13,20,12 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, -5, 9, 4, -3, -6, 8, -7, 5, -9, 6, -8, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uvw^{3}-uvw^{2}+uv+uw^{5}-uw^{4}+uw-u+vw^{5}-vw^{4}+vw-v-w^{5}+w^{3}-w^{2}}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$-q^{5}+q^{4}-q^{3}-q^{2}+q-1+4q^{-1}-2q^{-2}+4q^{-3}-q^{-4}+q^{-5}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$2z^{6}-3a^{2}z^{4}-z^{4}a^{-2}+12z^{4}+a^{4}z^{2}-14a^{2}z^{2}-6z^{2}a^{-2}-z^{2}a^{-4}+23z^{2}+4a^{4}-18a^{2}-7a^{-2}+21+3a^{4}z^{-2}-8a^{2}z^{-2}-2a^{-2}z^{-2}+7z^{-2}$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+a^{4}z^{8}+5a^{2}z^{8}+4z^{8}-4a^{3}z^{7}-az^{7}+3z^{7}a^{-1}-7a^{4}z^{6}-31a^{2}z^{6}-z^{6}a^{-2}+z^{6}a^{-4}-26z^{6}-2a^{3}z^{5}-21az^{5}-20z^{5}a^{-1}+z^{5}a^{-5}+18a^{4}z^{4}+61a^{2}z^{4}+3z^{4}a^{-2}-4z^{4}a^{-4}+50z^{4}+21a^{3}z^{3}+53az^{3}+33z^{3}a^{-1}-3z^{3}a^{-3}-4z^{3}a^{-5}-22a^{4}z^{2}-54a^{2}z^{2}-9z^{2}a^{-2}+z^{2}a^{-4}-42z^{2}-24a^{3}z-45az-21za^{-1}+3za^{-3}+3za^{-5}+13a^{4}+28a^{2}+7a^{-2}+a^{-4}+22+8a^{3}z^{-1}+15az^{-1}+7a^{-1}z^{-1}-a^{-3}z^{-1}-a^{-5}z^{-1}-3a^{4}z^{-2}-8a^{2}z^{-2}-2a^{-2}z^{-2}-7z^{-2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

0

7

1

0

5

3

1

-2

3

1

2

1

0

1

3

4

1

0

-1

1

3

-3

1

3

2

-5

3

1

2

-7

1

4

3

-9

0

-11

1

Integral Khovanov Homology

(db, data source)

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