L11n403

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(2)-1)(t(3)-1)\left(t(3)^{4}-t(1)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$-q^{5}+q^{-5}+q^{4}-q^{-4}+4q^{-3}-q^{2}-3q^{-2}+2q+4q^{-1}-2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}-2a^{2}z^{4}+5z^{4}+a^{4}z^{2}-8a^{2}z^{2}-z^{2}a^{-4}+8z^{2}+3a^{4}-10a^{2}-a^{-4}+8+2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}+4z^{-2}$ (db)
Kauffman polynomial	$z^{5}a^{-5}-4z^{3}a^{-5}+2za^{-5}+a^{4}z^{8}-7a^{4}z^{6}+z^{6}a^{-4}+18a^{4}z^{4}-5z^{4}a^{-4}-21a^{4}z^{2}+4z^{2}a^{-4}-2a^{4}z^{-2}+11a^{4}-a^{-4}+a^{3}z^{9}-4a^{3}z^{7}-a^{3}z^{5}-z^{5}a^{-3}+17a^{3}z^{3}+z^{3}a^{-3}-18a^{3}z-2za^{-3}+5a^{3}z^{-1}+a^{-3}z^{-1}+5a^{2}z^{8}-30a^{2}z^{6}+58a^{2}z^{4}-3z^{4}a^{-2}-51a^{2}z^{2}+z^{2}a^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}+24a^{2}+2a^{-2}+az^{9}+4z^{7}a^{-1}-24az^{5}-25z^{5}a^{-1}+52az^{3}+40z^{3}a^{-1}-37az-23za^{-1}+9az^{-1}+5a^{-1}z^{-1}+4z^{8}-24z^{6}+42z^{4}-33z^{2}-4z^{-2}+17$ (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

4

5

χ

11

1

-1

9

0

7

1

5

2

1

-1

3

2

1

3

4

1

0

-1

1

2

3

2

-3

2

3

1

-5

2

1

-7

1

4

3

-9

0

-11

1

Integral Khovanov Homology

(db, data source)

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