L11a194

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Link Presentations

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

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

Polynomial invariants

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

22

1

-1

20

2

18

3

1

-2

16

5

2

3

14

6

3

-3

12

6

5

1

10

6

0

8

5

6

-1

6

3

7

4

2

4

-2

2

1

4

3

0

1

-1

-2

1

Integral Khovanov Homology

(db, data source)

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