L11a81

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a81 at Knotilus!
	[edit L11a81 Quick Notes]

[edit L11a81 Further Notes and Views]

Link Presentations

[edit Notes on L11a81's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {5uv^{4}-11uv^{3}+12uv^{2}-7uv+2u+2v^{5}-7v^{4}+12v^{3}-11v^{2}+5v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {11}{q^{9/2}}}+{\frac {4}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {3}{q^{25/2}}}+{\frac {8}{q^{23/2}}}-{\frac {14}{q^{21/2}}}+{\frac {20}{q^{19/2}}}-{\frac {24}{q^{17/2}}}+{\frac {24}{q^{15/2}}}-{\frac {22}{q^{13/2}}}+{\frac {16}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}(-z)-a^{13}z^{-1}+3a^{11}z^{3}+4a^{11}z+a^{11}z^{-1}-2a^{9}z^{5}-a^{9}z^{3}+4a^{9}z+2a^{9}z^{-1}-4a^{7}z^{5}-11a^{7}z^{3}-9a^{7}z-2a^{7}z^{-1}-a^{5}z^{5}-a^{5}z^{3}$ (db)
Kauffman polynomial	$-z^{6}a^{16}+3z^{4}a^{16}-3z^{2}a^{16}+a^{16}-3z^{7}a^{15}+7z^{5}a^{15}-5z^{3}a^{15}+za^{15}-5z^{8}a^{14}+9z^{6}a^{14}-4z^{4}a^{14}-5z^{9}a^{13}+4z^{7}a^{13}+5z^{5}a^{13}-4z^{3}a^{13}-za^{13}+a^{13}z^{-1}-2z^{10}a^{12}-12z^{8}a^{12}+34z^{6}a^{12}-31z^{4}a^{12}+15z^{2}a^{12}-3a^{12}-13z^{9}a^{11}+18z^{7}a^{11}-3z^{5}a^{11}+2z^{3}a^{11}-4za^{11}+a^{11}z^{-1}-2z^{10}a^{10}-19z^{8}a^{10}+45z^{6}a^{10}-31z^{4}a^{10}+7z^{2}a^{10}-8z^{9}a^{9}+z^{7}a^{9}+17z^{5}a^{9}-15z^{3}a^{9}+7za^{9}-2a^{9}z^{-1}-12z^{8}a^{8}+17z^{6}a^{8}-4z^{4}a^{8}-5z^{2}a^{8}+3a^{8}-10z^{7}a^{7}+17z^{5}a^{7}-15z^{3}a^{7}+9za^{7}-2a^{7}z^{-1}-4z^{6}a^{6}+3z^{4}a^{6}-z^{5}a^{5}+z^{3}a^{5}$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

4

1

-3

-8

7

-10

9

4

-5

-12

13

7

6

-14

12

10

-2

-16

12

0

-18

8

12

4

-20

6

12

-6

-22

2

8

6

-24

1

6

-5

-26

2

-28

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$
$r=-11$	${\mathbb {Z} }$
$r=-10$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-9$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-8$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-7$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-6$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=-5$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=-4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{13}$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$