L11n357

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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_7,16,8,17 X_15,8,16,9 X_17,15,18,22 X_13,21,14,20 X_19,13,20,12 X_21,5,22,14 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, -6, 3, -8, 7, -9, 6}, {10, -1, -4, 5, 11, -2, -3, 8, -7, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(w-1)\left(-uv^{2}w-uv+u+v^{2}w^{3}-vw^{3}-w^{2}\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{-5}-q^{4}-q^{-4}+2q^{3}+3q^{-3}-2q^{2}-3q^{-2}+4q+4q^{-1}-3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$a^{4}z^{2}+a^{4}z^{-2}+3a^{4}-2a^{2}z^{4}-z^{4}a^{-2}-8a^{2}z^{2}-3z^{2}a^{-2}-2a^{2}z^{-2}-8a^{2}-a^{-2}+z^{6}+5z^{4}+8z^{2}+z^{-2}+6$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+a^{4}z^{8}+4a^{2}z^{8}+3z^{8}-5a^{3}z^{7}-2az^{7}+3z^{7}a^{-1}-7a^{4}z^{6}-24a^{2}z^{6}+2z^{6}a^{-2}-15z^{6}+5a^{3}z^{5}-8az^{5}-12z^{5}a^{-1}+z^{5}a^{-3}+17a^{4}z^{4}+47a^{2}z^{4}-6z^{4}a^{-2}+24z^{4}+5a^{3}z^{3}+18az^{3}+11z^{3}a^{-1}-2z^{3}a^{-3}-17a^{4}z^{2}-38a^{2}z^{2}+3z^{2}a^{-2}+2z^{2}a^{-4}-20z^{2}-7a^{3}z-10az-3za^{-1}+za^{-3}+za^{-5}+7a^{4}+14a^{2}-a^{-2}-a^{-4}+8+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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

9

1

-1

7

1

5

2

0

3

2

1

2

1

2

3

1

-1

2

1

-3

2

3

1

-5

1

0

-7

1

3

2

-9

0

-11

1

Integral Khovanov Homology

(db, data source)

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