L11a10

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_12,6,13,5 X₈₄₉₃ X_16,10,17,9 X_22,14,5,13 X_10,16,11,15 X_14,22,15,21 X_20,12,21,11 X_2,18,3,17
Gauss code	{1, -11, 5, -3}, {4, -1, 2, -5, 6, -8, 10, -4, 7, -9, 8, -6, 11, -2, 3, -10, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(4t(2)^{2}-7t(2)+4\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{17/2}+4q^{15/2}-7q^{13/2}+12q^{11/2}-16q^{9/2}+18q^{7/2}-20q^{5/2}+16q^{3/2}-13{\sqrt {q}}+{\frac {8}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {1}{q^{5/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-7}+a^{-7}z^{-1}+z^{5}a^{-5}-2za^{-5}-2a^{-5}z^{-1}+2z^{5}a^{-3}+3z^{3}a^{-3}+2za^{-3}+z^{5}a^{-1}-az^{3}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{7}a^{-9}-3z^{5}a^{-9}+2z^{3}a^{-9}+4z^{8}a^{-8}-15z^{6}a^{-8}+16z^{4}a^{-8}-3z^{2}a^{-8}-2a^{-8}+5z^{9}a^{-7}-16z^{7}a^{-7}+13z^{5}a^{-7}-2z^{3}a^{-7}+a^{-7}z^{-1}+2z^{10}a^{-6}+5z^{8}a^{-6}-32z^{6}a^{-6}+32z^{4}a^{-6}-3z^{2}a^{-6}-5a^{-6}+11z^{9}a^{-5}-27z^{7}a^{-5}+11z^{5}a^{-5}+5z^{3}a^{-5}-3za^{-5}+2a^{-5}z^{-1}+2z^{10}a^{-4}+11z^{8}a^{-4}-37z^{6}a^{-4}+25z^{4}a^{-4}-z^{2}a^{-4}-3a^{-4}+6z^{9}a^{-3}+z^{7}a^{-3}-24z^{5}a^{-3}+19z^{3}a^{-3}-4za^{-3}+10z^{8}a^{-2}-12z^{6}a^{-2}+a^{2}z^{4}+z^{4}a^{-2}+a^{-2}+11z^{7}a^{-1}+4az^{5}-15z^{5}a^{-1}-2az^{3}+8z^{3}a^{-1}-za^{-1}-a^{-1}z^{-1}+8z^{6}-7z^{4}+z^{2}$ (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

18

1

16

3

-3

14

4

1

3

12

8

3

-5

10

8

4

8

10

8

-2

6

10

8

2

4

6

10

4

2

7

10

-3

0

3

8

5

-2

1

5

-4

3

-6

1

-1

Integral Khovanov Homology

(db, data source)

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