L11n250

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n250 at Knotilus!
	[edit L11n250 Quick Notes]

[edit L11n250 Further Notes and Views]

Link Presentations

[edit Notes on L11n250's Link Presentations]

Planar diagram presentation	X_12,1,13,2 X_9,19,10,18 X_14,6,15,5 X_6,12,7,11 X_22,15,11,16 X_20,8,21,7 X₃₉₄₈ X_16,21,17,22 X_4,18,5,17 X_10,13,1,14 X_19,3,20,2
Gauss code	{1, 11, -7, -9, 3, -4, 6, 7, -2, -10}, {4, -1, 10, -3, 5, -8, 9, 2, -11, -6, 8, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{3}v^{3}-3u^{3}v^{2}+2u^{3}v-u^{2}v^{3}+4u^{2}v^{2}-2u^{2}v-2uv^{2}+4uv-u+2v^{2}-3v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{13/2}-4q^{11/2}+6q^{9/2}-8q^{7/2}+8q^{5/2}-9q^{3/2}+7{\sqrt {q}}-{\frac {5}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-2z^{5}a^{-1}+5z^{5}a^{-3}-z^{5}a^{-5}+az^{3}-7z^{3}a^{-1}+9z^{3}a^{-3}-4z^{3}a^{-5}+2az-5za^{-1}+8za^{-3}-4za^{-5}+za^{-7}+a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$z^{2}a^{-8}+4z^{3}a^{-7}-za^{-7}+2z^{6}a^{-6}-z^{4}a^{-6}+z^{2}a^{-6}+5z^{7}a^{-5}-14z^{5}a^{-5}+14z^{3}a^{-5}-6za^{-5}+a^{-5}z^{-1}+5z^{8}a^{-4}-15z^{6}a^{-4}+12z^{4}a^{-4}-3z^{2}a^{-4}-a^{-4}+2z^{9}a^{-3}-17z^{5}a^{-3}+18z^{3}a^{-3}-7za^{-3}+a^{-3}z^{-1}+8z^{8}a^{-2}-30z^{6}a^{-2}+29z^{4}a^{-2}-8z^{2}a^{-2}+2z^{9}a^{-1}+az^{7}-4z^{7}a^{-1}-4az^{5}-7z^{5}a^{-1}+5az^{3}+13z^{3}a^{-1}-2az-4za^{-1}+3z^{8}-13z^{6}+16z^{4}-5z^{2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

14

1

-1

12

3

10

4

2

-2

8

4

2

6

4

0

4

5

4

1

2

3

5

2

0

2

4

-2

1

3

2

-4

2

-2

-6

1

Integral Khovanov Homology

(db, data source)

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