L11a182

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Link Presentations

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2u^{2}v^{4}-5u^{2}v^{3}+6u^{2}v^{2}-4u^{2}v+u^{2}-3uv^{4}+9uv^{3}-11uv^{2}+9uv-3u+v^{4}-4v^{3}+6v^{2}-5v+2}{uv^{2}}}$ (db)
Jones polynomial	$23q^{9/2}-23q^{7/2}+19q^{5/2}-15q^{3/2}+{\frac {1}{q^{3/2}}}-q^{19/2}+4q^{17/2}-9q^{15/2}+15q^{13/2}-20q^{11/2}+8{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}+z^{7}a^{-5}-z^{5}a^{-1}+3z^{5}a^{-3}+3z^{5}a^{-5}-z^{5}a^{-7}-2z^{3}a^{-1}+3z^{3}a^{-3}+3z^{3}a^{-5}-2z^{3}a^{-7}+za^{-3}+za^{-5}-za^{-7}+a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-11}-z^{3}a^{-11}+4z^{6}a^{-10}-5z^{4}a^{-10}+z^{2}a^{-10}+8z^{7}a^{-9}-12z^{5}a^{-9}+6z^{3}a^{-9}-za^{-9}+10z^{8}a^{-8}-16z^{6}a^{-8}+11z^{4}a^{-8}-3z^{2}a^{-8}+7z^{9}a^{-7}-3z^{7}a^{-7}-8z^{5}a^{-7}+7z^{3}a^{-7}-za^{-7}+2z^{10}a^{-6}+15z^{8}a^{-6}-36z^{6}a^{-6}+26z^{4}a^{-6}-6z^{2}a^{-6}+12z^{9}a^{-5}-16z^{7}a^{-5}-z^{5}a^{-5}+5z^{3}a^{-5}-za^{-5}+2z^{10}a^{-4}+11z^{8}a^{-4}-28z^{6}a^{-4}+15z^{4}a^{-4}-2z^{2}a^{-4}+5z^{9}a^{-3}-z^{7}a^{-3}-16z^{5}a^{-3}+13z^{3}a^{-3}-2za^{-3}-a^{-3}z^{-1}+6z^{8}a^{-2}-11z^{6}a^{-2}+3z^{4}a^{-2}+z^{2}a^{-2}+a^{-2}+4z^{7}a^{-1}-10z^{5}a^{-1}+8z^{3}a^{-1}-za^{-1}-a^{-1}z^{-1}+z^{6}-2z^{4}+z^{2}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

7

8

χ

20

1

18

3

-3

16

6

1

5

14

9

3

-6

12

11

6

5

10

12

9

-3

8

11

0

6

9

13

4

6

10

-4

2

3

10

7

0

1

5

-4

-2

3

-4

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=2$	$i=4$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=3$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=4$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=7$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=8$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$