L11a387

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2uvw^{4}-3uvw^{3}+3uvw^{2}-2uvw+uv+uw^{5}-3uw^{4}+4uw^{3}-4uw^{2}+3uw-u+vw^{5}-3vw^{4}+4vw^{3}-4vw^{2}+3vw-v-w^{5}+2w^{4}-3w^{3}+3w^{2}-2w}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$q^{-3}-3q^{-4}+8q^{-5}-11q^{-6}+17q^{-7}-16q^{-8}+18q^{-9}-15q^{-10}+10q^{-11}-6q^{-12}+2q^{-13}-q^{-14}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$-2a^{14}z^{-2}-a^{14}+4a^{12}z^{2}+7a^{12}z^{-2}+12a^{12}-6a^{10}z^{4}-22a^{10}z^{2}-8a^{10}z^{-2}-24a^{10}+3a^{8}z^{6}+13a^{8}z^{4}+19a^{8}z^{2}+3a^{8}z^{-2}+13a^{8}+a^{6}z^{6}+3a^{6}z^{4}+2a^{6}z^{2}$ (db)
Kauffman polynomial	$a^{17}z^{5}-3a^{17}z^{3}+3a^{17}z-a^{17}z^{-1}+2a^{16}z^{6}-3a^{16}z^{4}+a^{16}+3a^{15}z^{7}-2a^{15}z^{5}-3a^{15}z^{3}+3a^{15}z-a^{15}z^{-1}+4a^{14}z^{8}-5a^{14}z^{6}+7a^{14}z^{4}-9a^{14}z^{2}-2a^{14}z^{-2}+7a^{14}+3a^{13}z^{9}+2a^{13}z^{7}-14a^{13}z^{5}+26a^{13}z^{3}-21a^{13}z+7a^{13}z^{-1}+a^{12}z^{10}+10a^{12}z^{8}-31a^{12}z^{6}+45a^{12}z^{4}-38a^{12}z^{2}-7a^{12}z^{-2}+22a^{12}+7a^{11}z^{9}-6a^{11}z^{7}-23a^{11}z^{5}+54a^{11}z^{3}-45a^{11}z+15a^{11}z^{-1}+a^{10}z^{10}+12a^{10}z^{8}-44a^{10}z^{6}+64a^{10}z^{4}-56a^{10}z^{2}-8a^{10}z^{-2}+28a^{10}+4a^{9}z^{9}-2a^{9}z^{7}-19a^{9}z^{5}+31a^{9}z^{3}-24a^{9}z+8a^{9}z^{-1}+6a^{8}z^{8}-19a^{8}z^{6}+26a^{8}z^{4}-25a^{8}z^{2}-3a^{8}z^{-2}+13a^{8}+3a^{7}z^{7}-7a^{7}z^{5}+3a^{7}z^{3}+a^{6}z^{6}-3a^{6}z^{4}+2a^{6}z^{2}$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-5

1

-7

3

1

-2

-9

5

-11

6

3

-3

-13

11

5

6

-15

8

9

1

-17

10

8

2

-19

5

8

3

-21

5

10

-5

-23

1

5

4

-25

1

5

-4

-27

1

-29

1

-1

Integral Khovanov Homology

(db, data source)

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