L11a483

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(w-1)^{2}\left(2vw^{2}-2vw+v+w^{3}-2w^{2}+2w\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$q^{9}-4q^{8}+9q^{7}-15q^{6}+22q^{5}-25q^{4}+27q^{3}-22q^{2}+18q-11+5q^{-1}-q^{-2}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{2}a^{-8}+a^{-8}-3z^{4}a^{-6}-6z^{2}a^{-6}+a^{-6}z^{-2}-3a^{-6}+2z^{6}a^{-4}+6z^{4}a^{-4}+8z^{2}a^{-4}-2a^{-4}z^{-2}+2a^{-4}+z^{6}a^{-2}-3z^{2}a^{-2}+a^{-2}z^{-2}-a^{-2}-z^{4}+1$ (db)
Kauffman polynomial	$2z^{10}a^{-4}+2z^{10}a^{-6}+9z^{9}a^{-3}+15z^{9}a^{-5}+6z^{9}a^{-7}+14z^{8}a^{-2}+26z^{8}a^{-4}+19z^{8}a^{-6}+7z^{8}a^{-8}+11z^{7}a^{-1}-17z^{7}a^{-5}-2z^{7}a^{-7}+4z^{7}a^{-9}-22z^{6}a^{-2}-70z^{6}a^{-4}-59z^{6}a^{-6}-15z^{6}a^{-8}+z^{6}a^{-10}+5z^{6}+az^{5}-14z^{5}a^{-1}-22z^{5}a^{-3}-18z^{5}a^{-5}-20z^{5}a^{-7}-9z^{5}a^{-9}+14z^{4}a^{-2}+65z^{4}a^{-4}+60z^{4}a^{-6}+11z^{4}a^{-8}-2z^{4}a^{-10}-4z^{4}+3z^{3}a^{-1}+15z^{3}a^{-3}+29z^{3}a^{-5}+23z^{3}a^{-7}+6z^{3}a^{-9}-9z^{2}a^{-2}-27z^{2}a^{-4}-25z^{2}a^{-6}-6z^{2}a^{-8}+z^{2}a^{-10}-za^{-1}-za^{-3}-5za^{-5}-7za^{-7}-2za^{-9}+a^{-2}+2a^{-4}+3a^{-6}+2a^{-8}+1-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-2}z^{-2}+2a^{-4}z^{-2}+a^{-6}z^{-2}$ (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

19

1

17

3

-3

15

6

1

5

13

9

3

-6

11

13

6

7

9

14

11

-3

7

13

11

2

5

9

14

5

3

9

13

-4

1

4

11

7

-1

1

7

-6

-3

4

-5

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=3$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=4$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{13}$
$r=5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=7$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=8$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$