L11a352

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X₈₄₉₃ X_14,5,15,6 X_20,7,21,8 X_16,13,17,14 X_18,10,19,9 X_10,11,1,12 X_6,15,7,16 X_4,21,5,22 X_22,17,11,18 X_2,20,3,19
Gauss code	{1, -11, 2, -9, 3, -8, 4, -2, 6, -7}, {7, -1, 5, -3, 8, -5, 10, -6, 11, -4, 9, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2u^{3}v^{3}-4u^{3}v^{2}+3u^{3}v-u^{3}-5u^{2}v^{3}+13u^{2}v^{2}-11u^{2}v+4u^{2}+4uv^{3}-11uv^{2}+13uv-5u-v^{3}+3v^{2}-4v+2}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	${\frac {28}{q^{9/2}}}-{\frac {28}{q^{7/2}}}+{\frac {23}{q^{5/2}}}+q^{3/2}-{\frac {17}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {5}{q^{17/2}}}-{\frac {12}{q^{15/2}}}+{\frac {19}{q^{13/2}}}-{\frac {25}{q^{11/2}}}-4{\sqrt {q}}+{\frac {9}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+z^{3}a^{7}-z^{7}a^{5}-2z^{5}a^{5}+2za^{5}+a^{5}z^{-1}-z^{7}a^{3}-3z^{5}a^{3}-5z^{3}a^{3}-5za^{3}-a^{3}z^{-1}+z^{5}a+2z^{3}a+za$ (db)
Kauffman polynomial	$a^{11}z^{5}+5a^{10}z^{6}-3a^{10}z^{4}+12a^{9}z^{7}-15a^{9}z^{5}+5a^{9}z^{3}-a^{9}z+16a^{8}z^{8}-24a^{8}z^{6}+11a^{8}z^{4}-3a^{8}z^{2}+11a^{7}z^{9}-3a^{7}z^{7}-20a^{7}z^{5}+13a^{7}z^{3}-2a^{7}z+3a^{6}z^{10}+24a^{6}z^{8}-58a^{6}z^{6}+36a^{6}z^{4}-6a^{6}z^{2}+18a^{5}z^{9}-24a^{5}z^{7}-8a^{5}z^{5}+19a^{5}z^{3}-7a^{5}z+a^{5}z^{-1}+3a^{4}z^{10}+15a^{4}z^{8}-43a^{4}z^{6}+31a^{4}z^{4}-5a^{4}z^{2}-a^{4}+7a^{3}z^{9}-5a^{3}z^{7}-13a^{3}z^{5}+18a^{3}z^{3}-8a^{3}z+a^{3}z^{-1}+7a^{2}z^{8}-13a^{2}z^{6}+7a^{2}z^{4}-a^{2}z^{2}+4az^{7}-9az^{5}+7az^{3}-2az+z^{6}-2z^{4}+z^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

3

0

6

1

-5

-2

11

3

8

-4

13

7

-6

15

10

5

-8

13

0

-10

12

15

-3

-12

8

14

6

-14

4

11

-7

-16

1

8

7

-18

4

-4

-20

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-5$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-4$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{12}$
$r=-3$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=-2$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{15}$	${\mathbb {Z} }^{15}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=0$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$