L11n405

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_16,9,17,10 X_8,15,9,16 X_4,17,1,18 X_22,12,19,11 X_10,4,11,3 X_5,21,6,20 X_21,5,22,18 X_12,20,13,19 X_2,14,3,13
Gauss code	{1, -11, 7, -5}, {10, 8, -9, -6}, {-8, -1, 2, -4, 3, -7, 6, -10, 11, -2, 4, -3, 5, 9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(w-1)^{2}\left(w^{2}+1\right)(vw+1)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$-q^{-5}+3q^{4}+3q^{-4}-4q^{3}-6q^{-3}+8q^{2}+8q^{-2}-9q-10q^{-1}+12$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-5z^{4}a^{-2}+13z^{4}-5a^{2}z^{2}-10z^{2}a^{-2}+z^{2}a^{-4}+14z^{2}-3a^{2}-11a^{-2}+3a^{-4}+11-a^{2}z^{-2}-5a^{-2}z^{-2}+2a^{-4}z^{-2}+4z^{-2}$ (db)
Kauffman polynomial	$a^{5}z^{5}-2a^{5}z^{3}+a^{5}z+3a^{4}z^{6}-6a^{4}z^{4}+6z^{4}a^{-4}+a^{4}z^{2}-17z^{2}a^{-4}-2a^{-4}z^{-2}+10a^{-4}+4a^{3}z^{7}+3z^{7}a^{-3}-7a^{3}z^{5}-9z^{5}a^{-3}+a^{3}z^{3}+14z^{3}a^{-3}-2a^{3}z-15za^{-3}+a^{3}z^{-1}+5a^{-3}z^{-1}+4a^{2}z^{8}+5z^{8}a^{-2}-9a^{2}z^{6}-22z^{6}a^{-2}+10a^{2}z^{4}+46z^{4}a^{-2}-9a^{2}z^{2}-45z^{2}a^{-2}-a^{2}z^{-2}-5a^{-2}z^{-2}+4a^{2}+22a^{-2}+2az^{9}+2z^{9}a^{-1}-z^{7}a^{-1}-11az^{5}-12z^{5}a^{-1}+23az^{3}+34z^{3}a^{-1}-17az-29za^{-1}+5az^{-1}+9a^{-1}z^{-1}+9z^{8}-34z^{6}+56z^{4}-38z^{2}-4z^{-2}+17$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

9

3

7

3

2

-1

5

1

4

3

4

3

-1

1

8

5

3

-1

4

6

2

-3

4

6

-2

-5

2

4

2

-7

1

4

-3

-9

2

-11

1

-1

Integral Khovanov Homology

(db, data source)

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