L11a388

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Link Presentations

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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_22,15,9,16 X_20,17,21,18 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, -7, 8, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {3uvw^{2}-4uvw+2uv+2uw^{3}-6uw^{2}+6uw-2u+2vw^{3}-6vw^{2}+6vw-2v-2w^{3}+4w^{2}-3w}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$q^{-2}-3q^{-3}+7q^{-4}-10q^{-5}+16q^{-6}-15q^{-7}+16q^{-8}-13q^{-9}+10q^{-10}-6q^{-11}+2q^{-12}-q^{-13}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}+3a^{12}z^{-2}+4a^{12}-6a^{10}z^{2}-2a^{10}z^{-2}-7a^{10}+3a^{8}z^{4}+3a^{8}z^{2}-a^{8}z^{-2}-a^{8}+3a^{6}z^{4}+5a^{6}z^{2}+a^{6}z^{-2}+4a^{6}+a^{4}z^{4}+a^{4}z^{2}$ (db)
Kauffman polynomial	$a^{15}z^{7}-5a^{15}z^{5}+9a^{15}z^{3}-7a^{15}z+2a^{15}z^{-1}+2a^{14}z^{8}-7a^{14}z^{6}+7a^{14}z^{4}-2a^{14}z^{2}-a^{14}z^{-2}+a^{14}+2a^{13}z^{9}-a^{13}z^{7}-18a^{13}z^{5}+36a^{13}z^{3}-27a^{13}z+8a^{13}z^{-1}+a^{12}z^{10}+5a^{12}z^{8}-22a^{12}z^{6}+19a^{12}z^{4}-7a^{12}z^{2}-3a^{12}z^{-2}+5a^{12}+6a^{11}z^{9}-6a^{11}z^{7}-27a^{11}z^{5}+49a^{11}z^{3}-34a^{11}z+10a^{11}z^{-1}+a^{10}z^{10}+10a^{10}z^{8}-28a^{10}z^{6}+17a^{10}z^{4}-5a^{10}z^{2}-2a^{10}z^{-2}+4a^{10}+4a^{9}z^{9}+3a^{9}z^{7}-23a^{9}z^{5}+23a^{9}z^{3}-10a^{9}z+2a^{9}z^{-1}+7a^{8}z^{8}-7a^{8}z^{6}-4a^{8}z^{4}+9a^{8}z^{2}+a^{8}z^{-2}-3a^{8}+7a^{7}z^{7}-6a^{7}z^{5}-a^{7}z^{3}+4a^{7}z-2a^{7}z^{-1}+6a^{6}z^{6}-8a^{6}z^{4}+8a^{6}z^{2}+a^{6}z^{-2}-4a^{6}+3a^{5}z^{5}-2a^{5}z^{3}+a^{4}z^{4}-a^{4}z^{2}$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-3

1

-5

3

1

-2

-7

4

-9

6

3

-3

-11

10

4

6

-13

8

9

1

-15

8

7

1

-17

5

8

3

-19

5

8

-3

-21

1

5

4

-23

1

5

-4

-25

1

-27

1

-1

Integral Khovanov Homology

(db, data source)

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