L11a197

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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_18,12,19,11 X_20,16,21,15 X_12,18,13,17 X_14,22,15,21 X₂₇₃₈ X_4,14,5,13 X_6,20,1,19
Gauss code	{1, -9, 2, -10, 4, -11}, {9, -1, 3, -2, 5, -7, 10, -8, 6, -4, 7, -5, 11, -6, 8, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{4}-t(2)^{4}+4t(1)^{2}t(2)^{3}-10t(1)t(2)^{3}+5t(2)^{3}-8t(1)^{2}t(2)^{2}+17t(1)t(2)^{2}-8t(2)^{2}+5t(1)^{2}t(2)-10t(1)t(2)+4t(2)-t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{21/2}-4q^{19/2}+9q^{17/2}-15q^{15/2}+21q^{13/2}-24q^{11/2}+24q^{9/2}-22q^{7/2}+15q^{5/2}-10q^{3/2}+4{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-3}-3z^{5}a^{-5}-z^{5}a^{-7}+z^{3}a^{-1}+2z^{3}a^{-3}-6z^{3}a^{-5}+z^{3}a^{-7}+z^{3}a^{-9}+5za^{-3}-6za^{-5}+2za^{-7}+2a^{-3}z^{-1}-3a^{-5}z^{-1}+a^{-7}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-12}-2z^{4}a^{-12}+z^{2}a^{-12}+4z^{7}a^{-11}-9z^{5}a^{-11}+6z^{3}a^{-11}-za^{-11}+7z^{8}a^{-10}-15z^{6}a^{-10}+10z^{4}a^{-10}-3z^{2}a^{-10}+6z^{9}a^{-9}-4z^{7}a^{-9}-12z^{5}a^{-9}+11z^{3}a^{-9}-3za^{-9}+2z^{10}a^{-8}+15z^{8}a^{-8}-41z^{6}a^{-8}+29z^{4}a^{-8}-6z^{2}a^{-8}-a^{-8}+13z^{9}a^{-7}-13z^{7}a^{-7}-18z^{5}a^{-7}+25z^{3}a^{-7}-8za^{-7}+a^{-7}z^{-1}+2z^{10}a^{-6}+19z^{8}a^{-6}-46z^{6}a^{-6}+31z^{4}a^{-6}-2z^{2}a^{-6}-3a^{-6}+7z^{9}a^{-5}+4z^{7}a^{-5}-31z^{5}a^{-5}+33z^{3}a^{-5}-13za^{-5}+3a^{-5}z^{-1}+11z^{8}a^{-4}-17z^{6}a^{-4}+10z^{4}a^{-4}-3a^{-4}+9z^{7}a^{-3}-15z^{5}a^{-3}+12z^{3}a^{-3}-7za^{-3}+2a^{-3}z^{-1}+4z^{6}a^{-2}-4z^{4}a^{-2}+z^{5}a^{-1}-z^{3}a^{-1}$ (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

3

18

6

1

-5

16

9

3

6

14

12

6

-6

12

9

3

10

12

0

8

10

12

-2

6

13

7

4

9

-5

2

1

7

6

0

3

-3

-2

1

Integral Khovanov Homology

(db, data source)

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