L11a245

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_16,7,17,8 X_2,15,3,16 X_18,5,19,6 X_12,3,13,4 X_22,11,7,12 X_4,21,5,22 X_14,20,15,19 X_20,14,21,13 X_6,9,1,10 X_10,17,11,18
Gauss code	{1, -3, 5, -7, 4, -10}, {2, -1, 10, -11, 6, -5, 9, -8, 3, -2, 11, -4, 8, -9, 7, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2t(1)t(2)^{4}-2t(2)^{4}+3t(1)^{2}t(2)^{3}-11t(1)t(2)^{3}+7t(2)^{3}-9t(1)^{2}t(2)^{2}+17t(1)t(2)^{2}-9t(2)^{2}+7t(1)^{2}t(2)-11t(1)t(2)+3t(2)-2t(1)^{2}+2t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {10}{q^{3/2}}}+{\frac {17}{q^{5/2}}}-{\frac {24}{q^{7/2}}}+{\frac {27}{q^{9/2}}}-{\frac {28}{q^{11/2}}}+{\frac {24}{q^{13/2}}}-{\frac {18}{q^{15/2}}}+{\frac {11}{q^{17/2}}}-{\frac {5}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)+a^{9}z+a^{7}z^{5}-2a^{7}z^{3}-2a^{7}z+a^{7}z^{-1}+3a^{5}z^{5}+4a^{5}z^{3}+a^{5}z-a^{5}z^{-1}+a^{3}z^{5}-2a^{3}z^{3}-3a^{3}z-az^{3}$ (db)
Kauffman polynomial	$a^{12}z^{6}-a^{12}z^{4}+5a^{11}z^{7}-9a^{11}z^{5}+3a^{11}z^{3}+a^{11}z+10a^{10}z^{8}-21a^{10}z^{6}+11a^{10}z^{4}-a^{10}z^{2}+10a^{9}z^{9}-16a^{9}z^{7}+3a^{9}z^{5}-a^{9}z^{3}+4a^{8}z^{10}+13a^{8}z^{8}-45a^{8}z^{6}+33a^{8}z^{4}-8a^{8}z^{2}+21a^{7}z^{9}-43a^{7}z^{7}+27a^{7}z^{5}-8a^{7}z^{3}+3a^{7}z-a^{7}z^{-1}+4a^{6}z^{10}+16a^{6}z^{8}-50a^{6}z^{6}+44a^{6}z^{4}-13a^{6}z^{2}+a^{6}+11a^{5}z^{9}-13a^{5}z^{7}+a^{5}z^{5}+5a^{5}z^{3}+a^{5}z-a^{5}z^{-1}+13a^{4}z^{8}-23a^{4}z^{6}+19a^{4}z^{4}-6a^{4}z^{2}+9a^{3}z^{7}-13a^{3}z^{5}+8a^{3}z^{3}-3a^{3}z+4a^{2}z^{6}-4a^{2}z^{4}+az^{5}-az^{3}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

3

-3

-2

7

1

6

-4

11

4

-7

-6

13

6

7

-8

14

11

-3

-10

14

13

1

-12

10

14

4

-14

8

14

-6

-16

4

11

7

-18

1

7

-6

-20

4

-22

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-6$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=-5$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-4$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=-3$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=-1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$