L11a434

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)(w-1)(v+w-1)(vw-v-w)}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{6}-4q^{5}+8q^{4}-14q^{3}+21q^{2}-22q+24-20q^{-1}+16q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}-z^{6}+2a^{2}z^{4}-2z^{4}a^{-2}+z^{4}a^{-4}-a^{4}z^{2}+a^{2}z^{2}-3z^{2}a^{-2}+z^{2}a^{-4}+2z^{2}+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$2z^{10}a^{-2}+2z^{10}+7az^{9}+13z^{9}a^{-1}+6z^{9}a^{-3}+10a^{2}z^{8}+14z^{8}a^{-2}+7z^{8}a^{-4}+17z^{8}+8a^{3}z^{7}-az^{7}-20z^{7}a^{-1}-7z^{7}a^{-3}+4z^{7}a^{-5}+4a^{4}z^{6}-14a^{2}z^{6}-46z^{6}a^{-2}-18z^{6}a^{-4}+z^{6}a^{-6}-45z^{6}+a^{5}z^{5}-11a^{3}z^{5}-12az^{5}+3z^{5}a^{-1}-7z^{5}a^{-3}-10z^{5}a^{-5}-5a^{4}z^{4}+8a^{2}z^{4}+48z^{4}a^{-2}+16z^{4}a^{-4}-2z^{4}a^{-6}+43z^{4}-a^{5}z^{3}+5a^{3}z^{3}+7az^{3}+5z^{3}a^{-1}+10z^{3}a^{-3}+6z^{3}a^{-5}+2a^{4}z^{2}-4a^{2}z^{2}-20z^{2}a^{-2}-6z^{2}a^{-4}-20z^{2}+1-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

13

1

11

3

-3

9

5

1

4

7

9

3

-6

5

12

5

7

3

10

9

-1

1

14

12

2

-1

10

14

4

-3

6

10

-4

-5

3

10

7

-7

1

6

-5

-9

3

-11

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{14}$
$r=1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$