L11n271

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_7,17,8,16 X_15,5,16,8 X_11,19,12,18 X_17,9,18,22 X_21,13,22,12 X_13,21,14,20 X_19,15,20,14 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, -5, 7, -8, 9, -4, 3, -6, 5, -9, 8, -7, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-4uvw^{2}+5uvw-2uv+2uw^{2}-2uw+2vw^{2}-2vw+2w^{3}-5w^{2}+4w}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{10}+2q^{9}-5q^{8}+7q^{7}-9q^{6}+11q^{5}-8q^{4}+9q^{3}-5q^{2}+3q$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-a^{-10}z^{-2}-a^{-10}+3z^{2}a^{-8}+3a^{-8}z^{-2}+5a^{-8}-2z^{4}a^{-6}-4z^{2}a^{-6}-2a^{-6}z^{-2}-5a^{-6}-2z^{4}a^{-4}-3z^{2}a^{-4}-a^{-4}z^{-2}-2a^{-4}+3z^{2}a^{-2}+a^{-2}z^{-2}+3a^{-2}$ (db)
Kauffman polynomial	$z^{7}a^{-11}-5z^{5}a^{-11}+9z^{3}a^{-11}-7za^{-11}+2a^{-11}z^{-1}+2z^{8}a^{-10}-8z^{6}a^{-10}+9z^{4}a^{-10}-2z^{2}a^{-10}-a^{-10}z^{-2}+a^{-10}+z^{9}a^{-9}+3z^{7}a^{-9}-26z^{5}a^{-9}+43z^{3}a^{-9}-27za^{-9}+8a^{-9}z^{-1}+7z^{8}a^{-8}-23z^{6}a^{-8}+20z^{4}a^{-8}-8z^{2}a^{-8}-3a^{-8}z^{-2}+5a^{-8}+z^{9}a^{-7}+10z^{7}a^{-7}-43z^{5}a^{-7}+52z^{3}a^{-7}-34za^{-7}+10a^{-7}z^{-1}+5z^{8}a^{-6}-8z^{6}a^{-6}-z^{4}a^{-6}-3z^{2}a^{-6}-2a^{-6}z^{-2}+4a^{-6}+8z^{7}a^{-5}-19z^{5}a^{-5}+18z^{3}a^{-5}-10za^{-5}+2a^{-5}z^{-1}+7z^{6}a^{-4}-12z^{4}a^{-4}+9z^{2}a^{-4}+a^{-4}z^{-2}-3a^{-4}+3z^{5}a^{-3}+4za^{-3}-2a^{-3}z^{-1}+6z^{2}a^{-2}+a^{-2}z^{-2}-4a^{-2}$ (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		\

0

1

2

3

4

5

6

7

8

9

χ

21

1

-1

19

1

17

4

1

-3

15

3

1

2

13

6

4

-2

11

5

3

2

9

4

7

3

7

5

4

1

5

4

3

5

-2

1

3

Integral Khovanov Homology

(db, data source)

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