L11n116

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_7,17,8,16 X_20,17,21,18 X_18,13,19,14 X_14,19,15,20 X_4,21,1,22 X_10,5,11,6 X_12,3,13,4 X_22,11,5,12 X_2,9,3,10 X_15,9,16,8
Gauss code	{1, -10, 8, -6}, {7, -1, -2, 11, 10, -7, 9, -8, 4, -5, -11, 2, 3, -4, 5, -3, 6, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)^{5}+2t(1)t(2)^{4}-4t(2)^{4}-7t(1)t(2)^{3}+7t(2)^{3}+7t(1)t(2)^{2}-7t(2)^{2}-4t(1)t(2)+2t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {3}{q^{3/2}}}+{\frac {6}{q^{5/2}}}-{\frac {11}{q^{7/2}}}+{\frac {13}{q^{9/2}}}-{\frac {14}{q^{11/2}}}+{\frac {14}{q^{13/2}}}-{\frac {11}{q^{15/2}}}+{\frac {7}{q^{17/2}}}-{\frac {4}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)+a^{9}z^{-1}+a^{7}z^{5}-3a^{7}z-3a^{7}z^{-1}+2a^{5}z^{5}+5a^{5}z^{3}+6a^{5}z+4a^{5}z^{-1}-3a^{3}z^{3}-5a^{3}z-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{6}-2a^{12}z^{4}+a^{12}z^{2}+4a^{11}z^{7}-11a^{11}z^{5}+8a^{11}z^{3}+5a^{10}z^{8}-11a^{10}z^{6}+4a^{10}z^{4}+a^{10}z^{2}-a^{10}+2a^{9}z^{9}+7a^{9}z^{7}-28a^{9}z^{5}+19a^{9}z^{3}-4a^{9}z+a^{9}z^{-1}+11a^{8}z^{8}-22a^{8}z^{6}+7a^{8}z^{4}+2a^{8}z^{2}-3a^{8}+2a^{7}z^{9}+10a^{7}z^{7}-31a^{7}z^{5}+27a^{7}z^{3}-12a^{7}z+3a^{7}z^{-1}+6a^{6}z^{8}-7a^{6}z^{6}+a^{6}z^{4}+5a^{6}z^{2}-3a^{6}+7a^{5}z^{7}-14a^{5}z^{5}+22a^{5}z^{3}-15a^{5}z+4a^{5}z^{-1}+3a^{4}z^{6}+3a^{4}z^{2}-2a^{4}+6a^{3}z^{3}-7a^{3}z+2a^{3}z^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

3

-4

4

1

-3

-6

7

2

5

-8

6

4

-2

-10

8

7

1

-12

7

0

-14

4

7

-3

-16

3

7

4

-18

1

4

-3

-20

3

-22

1

-1

Integral Khovanov Homology

(db, data source)

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