L11n274

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n274 at Knotilus!
	[edit L11n274 Quick Notes]

[edit L11n274 Further Notes and Views]

Link Presentations

[edit Notes on L11n274's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_13,2,14,3 X_14,7,15,8 X_9,18,10,19 X_17,11,18,22 X_11,21,12,20 X_21,17,22,16 X_4,15,1,16 X_19,10,20,5
Gauss code	{1, 4, -3, -10}, {-2, -1, 5, 3, -6, 11}, {-8, 2, -4, -5, 10, 9, -7, 6, -11, 8, -9, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{2}w^{4}-2uv^{2}w^{3}+2uv^{2}w^{2}+uvw^{3}-uvw^{2}+vw^{2}-vw-2w^{2}+2w-1}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q+2-3q^{-1}+4q^{-2}-4q^{-3}+5q^{-4}-3q^{-5}+4q^{-6}-q^{-7}+q^{-8}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{8}z^{2}+2a^{8}z^{-2}+2a^{8}-a^{6}z^{6}-6a^{6}z^{4}-12a^{6}z^{2}-5a^{6}z^{-2}-12a^{6}+a^{4}z^{8}+7a^{4}z^{6}+18a^{4}z^{4}+23a^{4}z^{2}+4a^{4}z^{-2}+14a^{4}-a^{2}z^{6}-5a^{2}z^{4}-7a^{2}z^{2}-a^{2}z^{-2}-4a^{2}$ (db)
Kauffman polynomial	$a^{10}z^{2}-a^{10}+a^{9}z^{3}+3a^{8}z^{4}-8a^{8}z^{2}-2a^{8}z^{-2}+8a^{8}+2a^{7}z^{7}-10a^{7}z^{5}+20a^{7}z^{3}-18a^{7}z+5a^{7}z^{-1}+3a^{6}z^{8}-16a^{6}z^{6}+31a^{6}z^{4}-33a^{6}z^{2}-5a^{6}z^{-2}+20a^{6}+a^{5}z^{9}-19a^{5}z^{5}+41a^{5}z^{3}-33a^{5}z+9a^{5}z^{-1}+5a^{4}z^{8}-26a^{4}z^{6}+42a^{4}z^{4}-32a^{4}z^{2}-4a^{4}z^{-2}+15a^{4}+a^{3}z^{9}-a^{3}z^{7}-14a^{3}z^{5}+29a^{3}z^{3}-19a^{3}z+5a^{3}z^{-1}+2a^{2}z^{8}-10a^{2}z^{6}+14a^{2}z^{4}-8a^{2}z^{2}-a^{2}z^{-2}+3a^{2}+az^{7}-5az^{5}+7az^{3}-4az+az^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

3

1

-1

1

-1

2

1

-1

-3

1

3

1

-5

3

0

-7

1

4

2

1

-9

1

2

3

2

-11

4

3

1

-13

1

2

3

-15

2

0

-17

1

Integral Khovanov Homology

(db, data source)

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