L11a88

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a88 at Knotilus!
	[edit L11a88 Quick Notes]

[edit L11a88 Further Notes and Views]

Link Presentations

[edit Notes on L11a88's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2uv^{3}-7uv^{2}+9uv-5u-5v^{3}+9v^{2}-7v+2}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-9q^{9/2}+13q^{7/2}-{\frac {1}{q^{7/2}}}-15q^{5/2}+{\frac {2}{q^{5/2}}}+14q^{3/2}-{\frac {6}{q^{3/2}}}+q^{15/2}-3q^{13/2}+6q^{11/2}-13{\sqrt {q}}+{\frac {9}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-7}-2z^{3}a^{-5}-za^{-5}+a^{-5}z^{-1}+z^{5}a^{-3}+a^{3}z-za^{-3}+a^{3}z^{-1}-2a^{-3}z^{-1}+z^{5}a^{-1}-2az^{3}-2az-az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-3z^{4}a^{-8}+z^{2}a^{-8}+3z^{7}a^{-7}-9z^{5}a^{-7}+5z^{3}a^{-7}-za^{-7}+5z^{8}a^{-6}-17z^{6}a^{-6}+18z^{4}a^{-6}-10z^{2}a^{-6}+3a^{-6}+4z^{9}a^{-5}-10z^{7}a^{-5}+5z^{5}a^{-5}+z^{3}a^{-5}+za^{-5}-a^{-5}z^{-1}+z^{10}a^{-4}+8z^{8}a^{-4}-38z^{6}a^{-4}+58z^{4}a^{-4}-34z^{2}a^{-4}+7a^{-4}+7z^{9}a^{-3}-21z^{7}a^{-3}+a^{3}z^{5}+30z^{5}a^{-3}-3a^{3}z^{3}-21z^{3}a^{-3}+3a^{3}z+9za^{-3}-a^{3}z^{-1}-2a^{-3}z^{-1}+z^{10}a^{-2}+6z^{8}a^{-2}+2a^{2}z^{6}-23z^{6}a^{-2}-3a^{2}z^{4}+37z^{4}a^{-2}-23z^{2}a^{-2}+a^{2}+4a^{-2}+3z^{9}a^{-1}+3az^{7}-5z^{7}a^{-1}-3az^{5}+12z^{5}a^{-1}-2az^{3}-16z^{3}a^{-1}+3az+7za^{-1}-az^{-1}-a^{-1}z^{-1}+3z^{8}-z^{6}-3z^{4}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

16

1

-1

14

2

12

4

1

-3

10

5

2

3

8

4

-4

6

7

5

2

4

7

8

1

2

6

7

-1

0

4

8

4

-2

2

5

-3

-4

4

-6

1

2

-1

-8

1

Integral Khovanov Homology

(db, data source)

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