L11n264

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n264 at Knotilus!
	[edit L11n264 Quick Notes]

[edit L11n264 Further Notes and Views]

Link Presentations

[edit Notes on L11n264's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_14,7,15,8 X_8,13,5,14 X_18,12,19,11 X_19,22,20,9 X_15,20,16,21 X_21,16,22,17 X_12,18,13,17 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, 5, -9, 4, -3, -7, 8, 9, -5, -6, 7, -8, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {-t(1)t(3)^{3}+t(1)t(2)t(3)^{3}-t(2)t(3)^{3}+t(3)^{3}+t(1)t(3)^{2}-2t(1)t(2)t(3)^{2}+t(2)t(3)^{2}-t(3)^{2}-t(1)t(3)+t(1)t(2)t(3)-t(2)t(3)+2t(3)+t(1)-t(1)t(2)+t(2)-1}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$1-3q^{-1}+5q^{-2}-5q^{-3}+7q^{-4}-5q^{-5}+6q^{-6}-3q^{-7}+q^{-8}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{8}z^{-2}+z^{4}a^{6}+z^{2}a^{6}-2a^{6}z^{-2}-3a^{6}-z^{6}a^{4}-3z^{4}a^{4}+a^{4}z^{-2}+3a^{4}+z^{4}a^{2}+2z^{2}a^{2}$ (db)
Kauffman polynomial	$2a^{9}z^{3}+a^{8}z^{8}-5a^{8}z^{6}+11a^{8}z^{4}-6a^{8}z^{2}-a^{8}z^{-2}+3a^{8}+a^{7}z^{9}-3a^{7}z^{7}+2a^{7}z^{5}+3a^{7}z^{3}-3a^{7}z+2a^{7}z^{-1}+4a^{6}z^{8}-16a^{6}z^{6}+22a^{6}z^{4}-14a^{6}z^{2}-2a^{6}z^{-2}+5a^{6}+a^{5}z^{9}-8a^{5}z^{5}+7a^{5}z^{3}-3a^{5}z+2a^{5}z^{-1}+3a^{4}z^{8}-10a^{4}z^{6}+8a^{4}z^{4}-6a^{4}z^{2}-a^{4}z^{-2}+3a^{4}+3a^{3}z^{7}-10a^{3}z^{5}+6a^{3}z^{3}+a^{2}z^{6}-3a^{2}z^{4}+2a^{2}z^{2}$ (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

χ

1

-1

2

-2

-3

3

1

2

-5

3

0

-7

4

2

-9

1

2

3

2

-11

5

4

1

-13

1

4

3

-15

2

-2

-17

1

Integral Khovanov Homology

(db, data source)

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