L11n259

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_7,15,8,14 X_13,5,14,8 X_11,18,12,19 X_22,20,9,19 X_20,16,21,15 X_16,22,17,21 X_17,12,18,13 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, -5, 9, -4, 3, 7, -8, -9, 5, 6, -7, 8, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uvw^{3}-2uvw^{2}+uvw-uv-uw^{3}+2uw^{2}-2uw+u-vw^{3}+2vw^{2}-2vw+v+w^{3}-w^{2}+2w-1}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{6}+4q^{5}-6q^{4}+8q^{3}-7q^{2}-q^{-2}+8q+4q^{-1}-5$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+3z^{4}a^{-2}-z^{4}a^{-4}-z^{4}+z^{2}a^{-2}-z^{2}a^{-4}-z^{2}-3a^{-2}+a^{-4}+2-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$z^{3}a^{-7}+4z^{4}a^{-6}-z^{2}a^{-6}+z^{7}a^{-5}+2z^{5}a^{-5}+z^{8}a^{-4}+2z^{6}a^{-4}-z^{4}a^{-4}-3z^{2}a^{-4}-a^{-4}z^{-2}+3a^{-4}+5z^{7}a^{-3}-6z^{5}a^{-3}+z^{3}a^{-3}-3za^{-3}+2a^{-3}z^{-1}+z^{8}a^{-2}+6z^{6}a^{-2}-14z^{4}a^{-2}+z^{2}a^{-2}-2a^{-2}z^{-2}+5a^{-2}+4z^{7}a^{-1}+az^{5}-7z^{5}a^{-1}-az^{3}+z^{3}a^{-1}-3za^{-1}+2a^{-1}z^{-1}+4z^{6}-9z^{4}+3z^{2}-z^{-2}+3$ (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

χ

13

1

-1

11

3

9

4

2

-2

7

4

2

5

3

4

1

3

5

4

1

3

6

3

-1

1

2

-1

-3

3

-5

1

-1

Integral Khovanov Homology

(db, data source)

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