L11n361

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_5,18,6,19 X₈₄₉₃ X_20,9,21,10 X_10,19,11,20 X_17,14,18,15 X_15,22,16,17 X_21,16,22,5 X_2,12,3,11
Gauss code	{1, -11, 5, -3}, {-8, 4, 7, -6, -10, 9}, {-4, -1, 2, -5, 6, -7, 11, -2, 3, 8, -9, 10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(w-1)^{2}\left(v^{2}+vw+w^{2}\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{-10}+3q^{-9}-4q^{-8}+7q^{-7}-7q^{-6}+8q^{-5}-7q^{-4}+6q^{-3}-3q^{-2}+2q^{-1}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{10}+3a^{8}z^{2}+a^{8}z^{-2}+4a^{8}-2a^{6}z^{4}-5a^{6}z^{2}-2a^{6}z^{-2}-6a^{6}-a^{4}z^{4}+a^{4}z^{-2}+a^{4}+2a^{2}z^{2}+2a^{2}$ (db)
Kauffman polynomial	$z^{7}a^{11}-4z^{5}a^{11}+4z^{3}a^{11}-za^{11}+3z^{8}a^{10}-14z^{6}a^{10}+19z^{4}a^{10}-8z^{2}a^{10}+2z^{9}a^{9}-5z^{7}a^{9}-5z^{5}a^{9}+12z^{3}a^{9}-2za^{9}+8z^{8}a^{8}-35z^{6}a^{8}+47z^{4}a^{8}-27z^{2}a^{8}-a^{8}z^{-2}+7a^{8}+2z^{9}a^{7}-2z^{7}a^{7}-12z^{5}a^{7}+17z^{3}a^{7}-9za^{7}+2a^{7}z^{-1}+5z^{8}a^{6}-19z^{6}a^{6}+28z^{4}a^{6}-25z^{2}a^{6}-2a^{6}z^{-2}+11a^{6}+4z^{7}a^{5}-10z^{5}a^{5}+10z^{3}a^{5}-7za^{5}+2a^{5}z^{-1}+2z^{6}a^{4}-3z^{2}a^{4}-a^{4}z^{-2}+3a^{4}+z^{5}a^{3}+z^{3}a^{3}+za^{3}+3z^{2}a^{2}-2a^{2}$ (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

2

-1

-5

3

-7

4

3

-1

-9

4

3

1

-11

3

4

1

-13

4

0

-15

2

5

3

-17

1

2

-1

-19

2

-21

1

-1

Integral Khovanov Homology

(db, data source)

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