L11n257

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,16,12,17 X_21,18,22,19 X_13,20,14,21 X_19,12,20,13 X_17,22,18,9 X_8,16,5,15 X_14,8,15,7 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 9, -8}, {11, -2, -3, 6, -5, -9, 8, 3, -7, 4, -6, 5, -4, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uvw^{2}-uvw-3uw^{2}+4uw-2u+2vw^{3}-4vw^{2}+3vw+w^{2}-w}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$2q^{-1}-2q^{-2}+6q^{-3}-6q^{-4}+8q^{-5}-6q^{-6}+6q^{-7}-5q^{-8}+2q^{-9}-q^{-10}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+3z^{2}a^{8}+3a^{8}z^{-2}+5a^{8}-2z^{4}a^{6}-5z^{2}a^{6}-2a^{6}z^{-2}-5a^{6}-z^{4}a^{4}-z^{2}a^{4}-a^{4}z^{-2}-2a^{4}+2z^{2}a^{2}+a^{2}z^{-2}+3a^{2}$ (db)
Kauffman polynomial	$a^{11}z^{7}-5a^{11}z^{5}+9a^{11}z^{3}-7a^{11}z+2a^{11}z^{-1}+2a^{10}z^{8}-8a^{10}z^{6}+8a^{10}z^{4}-2a^{10}z^{2}-a^{10}z^{-2}+a^{10}+a^{9}z^{9}+2a^{9}z^{7}-24a^{9}z^{5}+40a^{9}z^{3}-27a^{9}z+8a^{9}z^{-1}+6a^{8}z^{8}-23a^{8}z^{6}+22a^{8}z^{4}-7a^{8}z^{2}-3a^{8}z^{-2}+5a^{8}+a^{7}z^{9}+5a^{7}z^{7}-33a^{7}z^{5}+49a^{7}z^{3}-34a^{7}z+10a^{7}z^{-1}+4a^{6}z^{8}-13a^{6}z^{6}+11a^{6}z^{4}-4a^{6}z^{2}-2a^{6}z^{-2}+4a^{6}+4a^{5}z^{7}-13a^{5}z^{5}+17a^{5}z^{3}-10a^{5}z+2a^{5}z^{-1}+2a^{4}z^{6}-3a^{4}z^{4}+4a^{4}z^{2}+a^{4}z^{-2}-3a^{4}+a^{3}z^{5}-a^{3}z^{3}+4a^{3}z-2a^{3}z^{-1}+3a^{2}z^{2}+a^{2}z^{-2}-4a^{2}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-1

2

-3

3

0

-5

3

1

4

-7

4

0

-9

5

3

2

-11

2

5

1

2

-13

4

0

-15

1

2

1

-17

1

4

-3

-19

1

-21

1

-1

Integral Khovanov Homology

(db, data source)

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