L11a525

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_14,3,15,4 X_12,15,7,16 X_22,19,13,20 X_16,9,17,10 X_10,22,11,21 X_20,12,21,11 X_18,5,19,6 X₂₇₃₈ X_4,13,5,14 X_6,17,1,18
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 5, -6, 7, -3}, {10, -2, 3, -5, 11, -8, 4, -7, 6, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-u^{2}v^{2}w+u^{2}v^{2}-2u^{2}vw^{2}+5u^{2}vw-2u^{2}v+2u^{2}w^{2}-3u^{2}w+u^{2}-2uv^{2}w^{2}+3uv^{2}w-uv^{2}-uvw^{3}+6uvw^{2}-6uvw+uv+uw^{3}-3uw^{2}+2uw-v^{2}w^{3}+3v^{2}w^{2}-2v^{2}w+2vw^{3}-5vw^{2}+2vw-w^{3}+w^{2}}{uvw^{3/2}}}$ (db)
Jones polynomial	$q-3+7q^{-1}-12q^{-2}+17q^{-3}-18q^{-4}+20q^{-5}-16q^{-6}+13q^{-7}-8q^{-8}+4q^{-9}-q^{-10}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{10}+4z^{2}a^{8}+a^{8}z^{-2}+3a^{8}-3z^{4}a^{6}-4z^{2}a^{6}-2a^{6}z^{-2}-5a^{6}-3z^{4}a^{4}-z^{2}a^{4}+a^{4}z^{-2}+2a^{4}-z^{4}a^{2}+2z^{2}a^{2}+a^{2}+z^{2}$ (db)
Kauffman polynomial	$a^{11}z^{7}-3a^{11}z^{5}+3a^{11}z^{3}-a^{11}z+4a^{10}z^{8}-14a^{10}z^{6}+15a^{10}z^{4}-6a^{10}z^{2}+2a^{10}+5a^{9}z^{9}-13a^{9}z^{7}+2a^{9}z^{5}+10a^{9}z^{3}-3a^{9}z+2a^{8}z^{10}+8a^{8}z^{8}-46a^{8}z^{6}+59a^{8}z^{4}-34a^{8}z^{2}-a^{8}z^{-2}+10a^{8}+12a^{7}z^{9}-30a^{7}z^{7}+12a^{7}z^{5}+8a^{7}z^{3}-8a^{7}z+2a^{7}z^{-1}+2a^{6}z^{10}+14a^{6}z^{8}-56a^{6}z^{6}+66a^{6}z^{4}-41a^{6}z^{2}-2a^{6}z^{-2}+12a^{6}+7a^{5}z^{9}-7a^{5}z^{7}-9a^{5}z^{5}+12a^{5}z^{3}-6a^{5}z+2a^{5}z^{-1}+10a^{4}z^{8}-18a^{4}z^{6}+15a^{4}z^{4}-8a^{4}z^{2}-a^{4}z^{-2}+4a^{4}+9a^{3}z^{7}-13a^{3}z^{5}+9a^{3}z^{3}+6a^{2}z^{6}-6a^{2}z^{4}+4a^{2}z^{2}-a^{2}+3az^{5}-2az^{3}+z^{4}-z^{2}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

3

1

2

-2

-1

5

1

4

-3

8

3

-5

9

4

5

-7

10

9

-1

-9

10

8

2

-11

7

11

4

-13

6

9

-3

-15

3

8

5

-17

1

5

-4

-19

3

-21

1

-1

Integral Khovanov Homology

(db, data source)

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