L11a398

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a398 at Knotilus!
	[edit L11a398 Quick Notes]

[edit L11a398 Further Notes and Views]

Link Presentations

[edit Notes on L11a398's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_12,6,13,5 X₈₄₉₃ X_2,14,3,13 X_14,7,15,8 X_18,10,19,9 X_22,17,11,18 X_20,11,21,12 X_16,21,17,22 X_4,15,1,16 X_10,20,5,19
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	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)(t(3)-1)\left(t(2)t(3)^{3}-2t(2)t(3)^{2}+2t(3)^{2}+2t(2)t(3)-2t(3)+1\right)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$q^{-6}-q^{5}-5q^{-5}+4q^{4}+10q^{-4}-9q^{3}-17q^{-3}+17q^{2}+23q^{-2}-21q-25q^{-1}+27$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-6a^{2}z^{4}-3z^{4}a^{-2}+10z^{4}+a^{4}z^{2}-4a^{2}z^{2}-3z^{2}a^{-2}+6z^{2}-a^{4}+4a^{2}+a^{-2}-4-a^{4}z^{-2}+4a^{2}z^{-2}+2a^{-2}z^{-2}-5z^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-a^{6}z^{4}+5a^{5}z^{7}-10a^{5}z^{5}+z^{5}a^{-5}+5a^{5}z^{3}-z^{3}a^{-5}+a^{5}z-a^{5}z^{-1}+9a^{4}z^{8}-19a^{4}z^{6}+4z^{6}a^{-4}+12a^{4}z^{4}-5z^{4}a^{-4}-4a^{4}z^{2}+2z^{2}a^{-4}+a^{4}z^{-2}+7a^{3}z^{9}-a^{3}z^{7}+8z^{7}a^{-3}-25a^{3}z^{5}-10z^{5}a^{-3}+17a^{3}z^{3}+4z^{3}a^{-3}+5a^{3}z-5a^{3}z^{-1}+2a^{2}z^{10}+22a^{2}z^{8}+11z^{8}a^{-2}-63a^{2}z^{6}-17z^{6}a^{-2}+53a^{2}z^{4}+14z^{4}a^{-2}-18a^{2}z^{2}-5z^{2}a^{-2}+4a^{2}z^{-2}+2a^{-2}z^{-2}-2a^{2}-3a^{-2}+15az^{9}+8z^{9}a^{-1}-15az^{7}-z^{7}a^{-1}-19az^{5}-15z^{5}a^{-1}+17az^{3}+10z^{3}a^{-1}+9az+5za^{-1}-9az^{-1}-5a^{-1}z^{-1}+2z^{10}+24z^{8}-64z^{6}+59z^{4}-21z^{2}+5z^{-2}-4$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

3

7

6

1

-5

5

11

3

8

3

12

8

-4

1

15

9

6

-1

12

14

2

-3

11

13

-2

-5

6

12

6

-7

4

11

-7

-9

1

6

5

-11

4

-4

-13

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-3$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{15}$
$r=1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{11}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$