L11a16

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_14,6,15,5 X₈₄₉₃ X_18,10,19,9 X_20,12,21,11 X_22,14,5,13 X_10,20,11,19 X_12,22,13,21 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {4, -1, 2, -5, 6, -9, 7, -10, 8, -4, 11, -2, 3, -6, 9, -7, 10, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{6}-t(2)^{5}+t(2)^{4}-t(2)^{3}+t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{7/2}}}$ (db)
Jones polynomial	$-8q^{9/2}+6q^{7/2}-6q^{5/2}+3q^{3/2}-q^{21/2}+3q^{19/2}-4q^{17/2}+6q^{15/2}-7q^{13/2}+8q^{11/2}-3{\sqrt {q}}+{\frac {1}{\sqrt {q}}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$z^{9}a^{-5}-z^{7}a^{-3}+7z^{7}a^{-5}-z^{7}a^{-7}-5z^{5}a^{-3}+16z^{5}a^{-5}-5z^{5}a^{-7}-5z^{3}a^{-3}+12z^{3}a^{-5}-6z^{3}a^{-7}+3za^{-3}-3za^{-5}+3a^{-3}z^{-1}-5a^{-5}z^{-1}+2a^{-7}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-13}+3z^{4}a^{-12}-2z^{2}a^{-12}+4z^{5}a^{-11}-3z^{3}a^{-11}+4z^{6}a^{-10}-3z^{4}a^{-10}-2z^{2}a^{-10}+a^{-10}+4z^{7}a^{-9}-6z^{5}a^{-9}+4z^{8}a^{-8}-10z^{6}a^{-8}+4z^{4}a^{-8}+4z^{9}a^{-7}-16z^{7}a^{-7}+20z^{5}a^{-7}-12z^{3}a^{-7}+za^{-7}+2a^{-7}z^{-1}+2z^{10}a^{-6}-6z^{8}a^{-6}-z^{6}a^{-6}+6z^{4}a^{-6}+4z^{2}a^{-6}-5a^{-6}+7z^{9}a^{-5}-38z^{7}a^{-5}+63z^{5}a^{-5}-35z^{3}a^{-5}-za^{-5}+5a^{-5}z^{-1}+2z^{10}a^{-4}-9z^{8}a^{-4}+8z^{6}a^{-4}+2z^{4}a^{-4}+3z^{2}a^{-4}-5a^{-4}+3z^{9}a^{-3}-18z^{7}a^{-3}+33z^{5}a^{-3}-19z^{3}a^{-3}-2za^{-3}+3a^{-3}z^{-1}+z^{8}a^{-2}-5z^{6}a^{-2}+6z^{4}a^{-2}-z^{2}a^{-2}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

7

8

χ

22

1

20

2

-2

18

2

1

16

4

2

-2

14

3

2

1

12

5

4

-1

10

3

0

8

3

5

2

6

3

0

4

2

5

3

2

1

0

2

-2

1

-1

Integral Khovanov Homology

(db, data source)

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