L11a424

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a424 at Knotilus!
	[edit L11a424 Quick Notes]

[edit L11a424 Further Notes and Views]

Link Presentations

[edit Notes on L11a424's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_16,7,17,8 X_20,9,21,10 X_18,22,19,21 X_8,15,9,16 X_10,19,5,20 X_22,14,11,13 X_14,18,15,17 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 4, -7}, {11, -2, 8, -9, 6, -3, 9, -5, 7, -4, 5, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}+3t(1)t(2)^{2}-3t(1)t(3)t(2)^{2}+5t(3)t(2)^{2}-3t(2)^{2}-3t(1)t(3)^{2}t(2)+5t(3)^{2}t(2)-5t(1)t(2)+8t(1)t(3)t(2)-8t(3)t(2)+3t(2)+3t(1)t(3)^{2}-3t(3)^{2}+2t(1)-5t(1)t(3)+3t(3)-1}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{-9}-2q^{-8}+7q^{-7}-11q^{-6}+18q^{-5}-20q^{-4}+22q^{-3}-q^{2}-20q^{-2}+5q+15q^{-1}-10$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{10}z^{-2}-2a^{8}z^{-2}-4a^{8}+6a^{6}z^{2}+a^{6}z^{-2}+5a^{6}-4a^{4}z^{4}-5a^{4}z^{2}-2a^{4}+a^{2}z^{6}+2a^{2}z^{4}+4a^{2}z^{2}+a^{2}-z^{4}$ (db)
Kauffman polynomial	$a^{10}z^{6}-4a^{10}z^{4}+6a^{10}z^{2}+a^{10}z^{-2}-4a^{10}+2a^{9}z^{7}-4a^{9}z^{5}+4a^{9}z-2a^{9}z^{-1}+3a^{8}z^{8}-3a^{8}z^{6}-5a^{8}z^{4}+9a^{8}z^{2}+2a^{8}z^{-2}-6a^{8}+3a^{7}z^{9}-7a^{7}z^{5}+5a^{7}z^{3}-2a^{7}z^{-1}+a^{6}z^{10}+11a^{6}z^{8}-26a^{6}z^{6}+19a^{6}z^{4}-2a^{6}z^{2}+a^{6}z^{-2}-3a^{6}+8a^{5}z^{9}-a^{5}z^{7}-24a^{5}z^{5}+24a^{5}z^{3}-6a^{5}z+a^{4}z^{10}+18a^{4}z^{8}-37a^{4}z^{6}+21a^{4}z^{4}-2a^{4}z^{2}-a^{4}+5a^{3}z^{9}+11a^{3}z^{7}-37a^{3}z^{5}+24a^{3}z^{3}-2a^{3}z+10a^{2}z^{8}-10a^{2}z^{6}-4a^{2}z^{4}+3a^{2}z^{2}-a^{2}+10az^{7}-15az^{5}+z^{5}a^{-1}+5az^{3}+5z^{6}-5z^{4}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

5

1

-1

3

4

1

6

1

-5

-1

9

4

5

-3

12

7

-5

10

8

2

-7

10

12

2

-9

8

10

-2

-11

4

11

7

-13

3

7

-4

-15

1

6

5

-17

1

-1

-19

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{3}$
$r=-5$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$