L11a46

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_20,17,21,18 X_18,10,19,9 X_8,20,9,19 X_4,21,1,22 X_14,12,15,11 X_10,4,11,3 X_12,5,13,6 X_22,13,5,14 X_2,16,3,15
Gauss code	{1, -11, 8, -6}, {9, -1, 2, -5, 4, -8, 7, -9, 10, -7, 11, -2, 3, -4, 5, -3, 6, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)^{3}\left(t(2)^{2}-3t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {10}{q^{9/2}}}-5q^{7/2}+{\frac {16}{q^{7/2}}}+10q^{5/2}-{\frac {22}{q^{5/2}}}-17q^{3/2}+{\frac {26}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+22{\sqrt {q}}-{\frac {26}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z+a^{5}z^{-1}-2a^{3}z^{5}-4a^{3}z^{3}+z^{3}a^{-3}-4a^{3}z-2a^{3}z^{-1}-a^{-3}z^{-1}+az^{7}+3az^{5}-2z^{5}a^{-1}+5az^{3}-3z^{3}a^{-1}+3az+az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-a^{7}z^{3}+4a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+9a^{5}z^{7}-14a^{5}z^{5}+11a^{5}z^{3}-5a^{5}z+a^{5}z^{-1}+11a^{4}z^{8}-15a^{4}z^{6}+z^{6}a^{-4}+9a^{4}z^{4}-z^{4}a^{-4}-3a^{4}z^{2}+a^{4}+7a^{3}z^{9}+7a^{3}z^{7}+5z^{7}a^{-3}-35a^{3}z^{5}-10z^{5}a^{-3}+35a^{3}z^{3}+5z^{3}a^{-3}-15a^{3}z+za^{-3}+2a^{3}z^{-1}-a^{-3}z^{-1}+2a^{2}z^{10}+22a^{2}z^{8}+9z^{8}a^{-2}-51a^{2}z^{6}-19z^{6}a^{-2}+36a^{2}z^{4}+11z^{4}a^{-2}-12a^{2}z^{2}-2z^{2}a^{-2}+3a^{2}+a^{-2}+14az^{9}+7z^{9}a^{-1}-9az^{7}-2z^{7}a^{-1}-32az^{5}-22z^{5}a^{-1}+36az^{3}+18z^{3}a^{-1}-12az-za^{-1}+az^{-1}-a^{-1}z^{-1}+2z^{10}+20z^{8}-52z^{6}+35z^{4}-10z^{2}+2$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

4

6

1

-5

4

11

4

7

2

11

6

-5

0

15

11

4

-2

13

0

-4

9

13

-4

-6

7

13

6

-8

3

9

-6

-10

1

7

6

-12

3

-3

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-2$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{15}$
$r=1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$