L11n343

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,19,12,18 X_7,14,8,15 X_13,8,14,9 X_15,13,16,22 X_17,21,18,20 X_21,17,22,16 X_19,5,20,12 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 9}, {-5, 4, -6, 8, -7, 3, -9, 7, -8, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(v-1)(w-1)\left(uv-2u+2v^{2}-v\right)}{{\sqrt {u}}v^{3/2}{\sqrt {w}}}}$ (db)
Jones polynomial	$-2q^{3}+5q^{2}-6q+8-8q^{-1}+8q^{-2}-5q^{-3}+4q^{-4}-q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-2z^{2}a^{4}-2a^{4}z^{-2}-3a^{4}+z^{4}a^{2}+a^{2}z^{-2}+2z^{4}+4z^{2}+3-2z^{2}a^{-2}-a^{-2}$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+a^{4}z^{8}+4a^{2}z^{8}+3z^{8}+a^{5}z^{7}-2a^{3}z^{7}+3z^{7}a^{-1}+a^{6}z^{6}-12a^{2}z^{6}+z^{6}a^{-2}-10z^{6}-2a^{5}z^{5}+4a^{3}z^{5}-6z^{5}a^{-1}-5a^{6}z^{4}-9a^{4}z^{4}+15a^{2}z^{4}+4z^{4}a^{-2}+23z^{4}-3a^{5}z^{3}-10a^{3}z^{3}-az^{3}+9z^{3}a^{-1}+3z^{3}a^{-3}+8a^{6}z^{2}+13a^{4}z^{2}-8a^{2}z^{2}-7z^{2}a^{-2}-20z^{2}+6a^{5}z+8a^{3}z-4za^{-1}-2za^{-3}-5a^{6}-8a^{4}-a^{2}+2a^{-2}+5-2a^{5}z^{-1}-2a^{3}z^{-1}+a^{6}z^{-2}+2a^{4}z^{-2}+a^{2}z^{-2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

7

2

-2

5

3

2

-1

1

5

3

2

-1

5

0

-3

3

0

-5

2

5

3

-7

2

3

-1

-9

1

4

3

-11

0

-13

1

Integral Khovanov Homology

(db, data source)

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