L11a133

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a133 at Knotilus!
	[edit L11a133 Quick Notes]

[edit L11a133 Further Notes and Views]

Link Presentations

[edit Notes on L11a133's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_10,5,11,6 X_14,3,15,4 X_18,11,19,12 X_22,19,5,20 X_20,14,21,13 X_12,22,13,21 X_2,9,3,10 X_8,15,9,16
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -11, 10, -4, 6, -9, 8, -5, 11, -2, 3, -6, 7, -8, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)^{5}+4t(1)t(2)^{4}-6t(2)^{4}-14t(1)t(2)^{3}+14t(2)^{3}+14t(1)t(2)^{2}-14t(2)^{2}-6t(1)t(2)+4t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {25}{q^{9/2}}}-{\frac {22}{q^{7/2}}}+{\frac {15}{q^{5/2}}}-{\frac {10}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {5}{q^{19/2}}}+{\frac {10}{q^{17/2}}}-{\frac {16}{q^{15/2}}}+{\frac {22}{q^{13/2}}}-{\frac {25}{q^{11/2}}}-{\sqrt {q}}+{\frac {4}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{9}+za^{9}+a^{9}z^{-1}+z^{5}a^{7}-2z^{3}a^{7}-5za^{7}-3a^{7}z^{-1}+3z^{5}a^{5}+6z^{3}a^{5}+7za^{5}+4a^{5}z^{-1}+z^{5}a^{3}-2z^{3}a^{3}-5za^{3}-2a^{3}z^{-1}-z^{3}a$ (db)
Kauffman polynomial	$-z^{6}a^{12}+z^{4}a^{12}-5z^{7}a^{11}+10z^{5}a^{11}-4z^{3}a^{11}-za^{11}-9z^{8}a^{10}+20z^{6}a^{10}-12z^{4}a^{10}+z^{2}a^{10}+a^{10}-7z^{9}a^{9}+4z^{7}a^{9}+17z^{5}a^{9}-14z^{3}a^{9}+3za^{9}-a^{9}z^{-1}-2z^{10}a^{8}-19z^{8}a^{8}+52z^{6}a^{8}-36z^{4}a^{8}+5z^{2}a^{8}+3a^{8}-14z^{9}a^{7}+13z^{7}a^{7}+24z^{5}a^{7}-33z^{3}a^{7}+13za^{7}-3a^{7}z^{-1}-2z^{10}a^{6}-21z^{8}a^{6}+52z^{6}a^{6}-38z^{4}a^{6}+5z^{2}a^{6}+3a^{6}-7z^{9}a^{5}-5z^{7}a^{5}+33z^{5}a^{5}-36z^{3}a^{5}+16za^{5}-4a^{5}z^{-1}-11z^{8}a^{4}+17z^{6}a^{4}-11z^{4}a^{4}+z^{2}a^{4}+2a^{4}-9z^{7}a^{3}+15z^{5}a^{3}-12z^{3}a^{3}+7za^{3}-2a^{3}z^{-1}-4z^{6}a^{2}+4z^{4}a^{2}-z^{5}a+z^{3}a$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

3

-3

-2

7

1

6

-4

9

4

-5

-6

13

6

7

-8

12

9

-3

-10

13

0

-12

10

13

3

-14

6

12

-6

-16

4

10

6

-18

1

6

-5

-20

4

-22

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-6$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-5$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-4$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{13}$
$r=-3$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=-1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$