L11a404

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_10,13,5,14 X_8,17,9,18 X_14,7,15,8 X_18,9,19,10 X_22,20,11,19 X_20,16,21,15 X_16,22,17,21 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 5, -4, 6, -3}, {11, -2, 3, -5, 8, -9, 4, -6, 7, -8, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{2}w^{3}-2uv^{2}w^{2}+2uv^{2}w+uvw^{4}-4uvw^{3}+7uvw^{2}-5uvw+uv+2uw^{3}-4uw^{2}+3uw-u+v^{2}w^{4}-3v^{2}w^{3}+4v^{2}w^{2}-2v^{2}w-vw^{4}+5vw^{3}-7vw^{2}+4vw-v-2w^{3}+2w^{2}-w}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{-9}-3q^{-8}+8q^{-7}-12q^{-6}+19q^{-5}-20q^{-4}+22q^{-3}-q^{2}-19q^{-2}+4q+14q^{-1}-9$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{2}+2a^{8}z^{-2}+2a^{8}-3a^{6}z^{4}-9a^{6}z^{2}-5a^{6}z^{-2}-12a^{6}+2a^{4}z^{6}+8a^{4}z^{4}+16a^{4}z^{2}+4a^{4}z^{-2}+14a^{4}+a^{2}z^{6}+a^{2}z^{4}-2a^{2}z^{2}-a^{2}z^{-2}-4a^{2}-z^{4}-z^{2}$ (db)
Kauffman polynomial	$z^{6}a^{10}-3z^{4}a^{10}+3z^{2}a^{10}-a^{10}+3z^{7}a^{9}-7z^{5}a^{9}+4z^{3}a^{9}+5z^{8}a^{8}-11z^{6}a^{8}+10z^{4}a^{8}-11z^{2}a^{8}-2a^{8}z^{-2}+8a^{8}+4z^{9}a^{7}-18z^{5}a^{7}+25z^{3}a^{7}-18za^{7}+5a^{7}z^{-1}+z^{10}a^{6}+16z^{8}a^{6}-50z^{6}a^{6}+62z^{4}a^{6}-45z^{2}a^{6}-5a^{6}z^{-2}+20a^{6}+9z^{9}a^{5}-3z^{7}a^{5}-34z^{5}a^{5}+53z^{3}a^{5}-33za^{5}+9a^{5}z^{-1}+z^{10}a^{4}+20z^{8}a^{4}-55z^{6}a^{4}+62z^{4}a^{4}-38z^{2}a^{4}-4a^{4}z^{-2}+15a^{4}+5z^{9}a^{3}+8z^{7}a^{3}-37z^{5}a^{3}+41z^{3}a^{3}-19za^{3}+5a^{3}z^{-1}+9z^{8}a^{2}-13z^{6}a^{2}+8z^{4}a^{2}-6z^{2}a^{2}-a^{2}z^{-2}+3a^{2}+8z^{7}a-13z^{5}a+8z^{3}a-4za+az^{-1}+4z^{6}-5z^{4}+z^{2}+z^{5}a^{-1}-z^{3}a^{-1}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

5

1

-1

3

1

6

1

-5

-1

8

3

5

-3

12

7

-5

10

7

3

-7

10

12

2

-9

9

10

-1

-11

4

11

7

-13

4

8

-4

-15

1

6

5

-17

2

-2

-19

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{4}$
$r=-5$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$