L11n258

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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_19,22,20,9 X_15,20,16,21 X_21,16,22,17 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 $u$ , $v$ , $w$ , ...)	${\frac {uvw^{3}-4uvw^{2}+5uvw-3uv-uw^{3}+2uw^{2}-2uw+u-vw^{3}+2vw^{2}-2vw+v+3w^{3}-5w^{2}+4w-1}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-3q^{-6}+6q^{-5}-10q^{-4}+q^{3}+13q^{-3}-2q^{2}-12q^{-2}+7q+13q^{-1}-9$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{6}z^{-2}-2a^{6}-z^{4}a^{4}+2z^{2}a^{4}+3a^{4}z^{-2}+6a^{4}+z^{6}a^{2}+2z^{4}a^{2}-2a^{2}z^{-2}-3a^{2}-2z^{4}-4z^{2}-z^{-2}-3+z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$6a^{7}z^{3}-7a^{7}z+2a^{7}z^{-1}+3a^{6}z^{6}+a^{6}z^{2}-a^{6}z^{-2}+a^{6}+7a^{5}z^{7}-17a^{5}z^{5}+32a^{5}z^{3}-27a^{5}z+8a^{5}z^{-1}+5a^{4}z^{8}-4a^{4}z^{6}-2a^{4}z^{4}+a^{4}z^{2}-3a^{4}z^{-2}+5a^{4}+a^{3}z^{9}+14a^{3}z^{7}-43a^{3}z^{5}+51a^{3}z^{3}-34a^{3}z+10a^{3}z^{-1}+8a^{2}z^{8}-13a^{2}z^{6}+z^{6}a^{-2}+a^{2}z^{4}-4z^{4}a^{-2}-2a^{2}z^{2}+6z^{2}a^{-2}-2a^{2}z^{-2}+a^{-2}z^{-2}+4a^{2}-4a^{-2}+az^{9}+9az^{7}+2z^{7}a^{-1}-30az^{5}-4z^{5}a^{-1}+25az^{3}-10az+4za^{-1}+2az^{-1}-2a^{-1}z^{-1}+3z^{8}-5z^{6}-z^{4}+4z^{2}+z^{-2}-3$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

2

1

-1

3

5

1

4

2

-2

-1

9

5

4

-3

6

7

1

-5

7

6

1

-7

3

6

3

-9

3

7

-4

-11

3

-13

3

-3

Integral Khovanov Homology

(db, data source)

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