L11a306

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_4,9,5,10 X_16,5,17,6 X_22,17,9,18 X_18,13,19,14 X_14,21,15,22 X_20,7,21,8 X_6,15,7,16 X_8,19,1,20
Gauss code	{1, -2, 3, -4, 5, -10, 9, -11}, {4, -1, 2, -3, 7, -8, 10, -5, 6, -7, 11, -9, 8, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)t(2)^{5}-3t(1)^{2}t(2)^{4}+4t(1)t(2)^{4}-t(2)^{4}-3t(1)^{3}t(2)^{3}+7t(1)^{2}t(2)^{3}-8t(1)t(2)^{3}+3t(2)^{3}+3t(1)^{3}t(2)^{2}-8t(1)^{2}t(2)^{2}+7t(1)t(2)^{2}-3t(2)^{2}-t(1)^{3}t(2)+4t(1)^{2}t(2)-3t(1)t(2)-t(1)^{2}}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {4}{q^{25/2}}}+{\frac {8}{q^{23/2}}}-{\frac {13}{q^{21/2}}}+{\frac {17}{q^{19/2}}}-{\frac {19}{q^{17/2}}}+{\frac {19}{q^{15/2}}}-{\frac {17}{q^{13/2}}}+{\frac {11}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}(-z)-a^{13}z^{-1}+4a^{11}z^{3}+8a^{11}z+3a^{11}z^{-1}-3a^{9}z^{5}-8a^{9}z^{3}-6a^{9}z-2a^{9}z^{-1}-3a^{7}z^{5}-8a^{7}z^{3}-5a^{7}z-a^{5}z^{5}-2a^{5}z^{3}$ (db)
Kauffman polynomial	$a^{16}z^{6}-2a^{16}z^{4}+a^{16}z^{2}+4a^{15}z^{7}-10a^{15}z^{5}+6a^{15}z^{3}+6a^{14}z^{8}-13a^{14}z^{6}+5a^{14}z^{4}-a^{14}z^{2}+a^{14}+5a^{13}z^{9}-6a^{13}z^{7}-5a^{13}z^{5}+2a^{13}z^{3}+a^{13}z-a^{13}z^{-1}+2a^{12}z^{10}+7a^{12}z^{8}-27a^{12}z^{6}+30a^{12}z^{4}-19a^{12}z^{2}+3a^{12}+11a^{11}z^{9}-28a^{11}z^{7}+34a^{11}z^{5}-27a^{11}z^{3}+14a^{11}z-3a^{11}z^{-1}+2a^{10}z^{10}+8a^{10}z^{8}-31a^{10}z^{6}+42a^{10}z^{4}-20a^{10}z^{2}+3a^{10}+6a^{9}z^{9}-12a^{9}z^{7}+14a^{9}z^{5}-8a^{9}z^{3}+8a^{9}z-2a^{9}z^{-1}+7a^{8}z^{8}-15a^{8}z^{6}+14a^{8}z^{4}-3a^{8}z^{2}+6a^{7}z^{7}-14a^{7}z^{5}+13a^{7}z^{3}-5a^{7}z+3a^{6}z^{6}-5a^{6}z^{4}+a^{5}z^{5}-2a^{5}z^{3}$ (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

-4

1

-6

3

1

-2

-8

4

-10

7

3

-4

-12

10

4

6

-14

9

7

-2

-16

10

0

-18

7

9

2

-20

6

10

-4

-22

3

8

5

-24

1

5

-4

-26

3

-28

1

-1

Integral Khovanov Homology

(db, data source)

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