L11a17

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{4}-3t(2)^{3}+3t(2)^{2}-3t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{19/2}+4q^{17/2}-7q^{15/2}+12q^{13/2}-15q^{11/2}+16q^{9/2}-17q^{7/2}+13q^{5/2}-10q^{3/2}+5{\sqrt {q}}-{\frac {3}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}+z^{7}a^{-5}-z^{5}a^{-1}+4z^{5}a^{-3}+3z^{5}a^{-5}-z^{5}a^{-7}-3z^{3}a^{-1}+6z^{3}a^{-3}-2z^{3}a^{-7}-za^{-1}+6za^{-3}-6za^{-5}+za^{-7}+3a^{-3}z^{-1}-5a^{-5}z^{-1}+2a^{-7}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-4}-2z^{10}a^{-6}-4z^{9}a^{-3}-9z^{9}a^{-5}-5z^{9}a^{-7}-4z^{8}a^{-2}-2z^{8}a^{-6}-6z^{8}a^{-8}-3z^{7}a^{-1}+10z^{7}a^{-3}+28z^{7}a^{-5}+9z^{7}a^{-7}-6z^{7}a^{-9}+11z^{6}a^{-2}+10z^{6}a^{-4}+9z^{6}a^{-6}+7z^{6}a^{-8}-4z^{6}a^{-10}-z^{6}+10z^{5}a^{-1}-5z^{5}a^{-3}-37z^{5}a^{-5}-12z^{5}a^{-7}+9z^{5}a^{-9}-z^{5}a^{-11}-6z^{4}a^{-2}-13z^{4}a^{-4}-9z^{4}a^{-6}+2z^{4}a^{-8}+7z^{4}a^{-10}+3z^{4}-8z^{3}a^{-1}-6z^{3}a^{-3}+13z^{3}a^{-5}+9z^{3}a^{-7}-z^{3}a^{-9}+z^{3}a^{-11}-2z^{2}a^{-4}-3z^{2}a^{-6}-2z^{2}a^{-8}-2z^{2}a^{-10}-z^{2}+2za^{-1}+8za^{-3}+7za^{-5}+za^{-7}+5a^{-4}+5a^{-6}-a^{-10}-3a^{-3}z^{-1}-5a^{-5}z^{-1}-2a^{-7}z^{-1}$ (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

20

1

18

3

-3

16

4

1

3

14

8

3

-5

12

7

4

3

10

9

8

-1

8

7

1

6

5

9

4

5

8

-3

2

7

5

0

1

3

-2

2

-4

1

-1

Integral Khovanov Homology

(db, data source)

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