L11a472

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(w-1)^{2}\left(v^{2}w^{2}-v^{2}w-vw^{2}+vw-v-w+1\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{7}-4q^{6}+8q^{5}-12q^{4}-q^{-4}+17q^{3}+3q^{-3}-17q^{2}-6q^{-2}+18q+11q^{-1}-14$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-2}-5z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-9z^{4}a^{-2}+3z^{4}a^{-4}+8z^{4}-3a^{2}z^{2}-8z^{2}a^{-2}+2z^{2}a^{-4}+9z^{2}-a^{2}-4a^{-2}+a^{-4}+4-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-8}+4z^{5}a^{-7}-2z^{3}a^{-7}+8z^{6}a^{-6}-8z^{4}a^{-6}+2z^{2}a^{-6}+10z^{7}a^{-5}-11z^{5}a^{-5}+z^{3}a^{-5}+10z^{8}a^{-4}-15z^{6}a^{-4}+6z^{4}a^{-4}-3z^{2}a^{-4}-a^{-4}z^{-2}+3a^{-4}+7z^{9}a^{-3}+a^{3}z^{7}-10z^{7}a^{-3}-4a^{3}z^{5}+z^{5}a^{-3}+4a^{3}z^{3}+z^{3}a^{-3}-a^{3}z-2za^{-3}+2a^{-3}z^{-1}+2z^{10}a^{-2}+3a^{2}z^{8}+10z^{8}a^{-2}-12a^{2}z^{6}-43z^{6}a^{-2}+15a^{2}z^{4}+48z^{4}a^{-2}-9a^{2}z^{2}-23z^{2}a^{-2}-2a^{-2}z^{-2}+2a^{2}+7a^{-2}+4az^{9}+11z^{9}a^{-1}-13az^{7}-34z^{7}a^{-1}+9az^{5}+29z^{5}a^{-1}+2az^{3}-4z^{3}a^{-1}-3az-4za^{-1}+2a^{-1}z^{-1}+2z^{10}+3z^{8}-32z^{6}+48z^{4}-27z^{2}-z^{-2}+7$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

15

1

13

3

-3

11

5

1

4

9

7

3

-4

7

10

5

9

0

3

9

8

1

7

11

4

-1

4

7

-3

2

7

5

-5

1

4

-3

-7

2

-9

1

-1

Integral Khovanov Homology

(db, data source)

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