L11a215

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_12,4,13,3 X_22,12,7,11 X_20,16,21,15 X_18,10,19,9 X_10,20,11,19 X_14,22,15,21 X_16,6,17,5 X₂₇₃₈ X_4,14,5,13 X_6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 5, -6, 3, -2, 10, -7, 4, -8, 11, -5, 6, -4, 7, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2u^{2}v^{4}-4u^{2}v^{3}+5u^{2}v^{2}-4u^{2}v+u^{2}-3uv^{4}+6uv^{3}-7uv^{2}+6uv-3u+v^{4}-4v^{3}+5v^{2}-4v+2}{uv^{2}}}$ (db)
Jones polynomial	$-16q^{9/2}+11q^{7/2}-8q^{5/2}+3q^{3/2}+q^{23/2}-3q^{21/2}+7q^{19/2}-12q^{17/2}+16q^{15/2}-18q^{13/2}+18q^{11/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-5}-z^{7}a^{-7}+z^{5}a^{-3}-3z^{5}a^{-5}-4z^{5}a^{-7}+z^{5}a^{-9}+3z^{3}a^{-3}-z^{3}a^{-5}-7z^{3}a^{-7}+3z^{3}a^{-9}+3za^{-3}+2za^{-5}-7za^{-7}+3za^{-9}+a^{-3}z^{-1}-2a^{-7}z^{-1}+a^{-9}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-6}-2z^{10}a^{-8}-4z^{9}a^{-5}-11z^{9}a^{-7}-7z^{9}a^{-9}-3z^{8}a^{-4}-4z^{8}a^{-6}-11z^{8}a^{-8}-10z^{8}a^{-10}-z^{7}a^{-3}+10z^{7}a^{-5}+30z^{7}a^{-7}+10z^{7}a^{-9}-9z^{7}a^{-11}+10z^{6}a^{-4}+29z^{6}a^{-6}+45z^{6}a^{-8}+20z^{6}a^{-10}-6z^{6}a^{-12}+4z^{5}a^{-3}-z^{5}a^{-5}-22z^{5}a^{-7}+14z^{5}a^{-11}-3z^{5}a^{-13}-9z^{4}a^{-4}-33z^{4}a^{-6}-50z^{4}a^{-8}-19z^{4}a^{-10}+6z^{4}a^{-12}-z^{4}a^{-14}-6z^{3}a^{-3}-6z^{3}a^{-5}+11z^{3}a^{-7}-2z^{3}a^{-9}-11z^{3}a^{-11}+2z^{3}a^{-13}+z^{2}a^{-4}+15z^{2}a^{-6}+27z^{2}a^{-8}+9z^{2}a^{-10}-3z^{2}a^{-12}+z^{2}a^{-14}+4za^{-3}-7za^{-7}-za^{-9}+2za^{-11}+a^{-4}-3a^{-6}-5a^{-8}-2a^{-10}-a^{-3}z^{-1}+2a^{-7}z^{-1}+a^{-9}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

24

1

-1

22

2

20

5

1

-4

18

7

2

5

16

9

5

-4

14

9

7

2

12

9

0

10

7

9

-2

8

5

10

5

6

3

6

-3

4

1

6

5

2

-2

0

1

Integral Khovanov Homology

(db, data source)

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