L11n394

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_9,22,10,19 X₃₈₄₉ X_21,17,22,16 X_11,5,12,18 X_5,21,6,20 X_17,11,18,10 X_19,12,20,13 X_13,2,14,3
Gauss code	{1, 11, -5, -3}, {-10, 8, -6, 4}, {-8, -1, 2, 5, -4, 9, -7, 10, -11, -2, 3, 6, -9, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(v-1)(w-1)\left(w^{2}-w+1\right)\left(uw^{2}-1\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$-q^{-6}+4q^{-5}-4q^{-4}+q^{3}+8q^{-3}-3q^{2}-8q^{-2}+5q+8q^{-1}-6$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+z^{6}+5a^{4}z^{4}-12a^{2}z^{4}+4z^{4}-a^{6}z^{2}+7a^{4}z^{2}-10a^{2}z^{2}+4z^{2}-a^{6}+2a^{4}-3a^{2}+2+a^{6}z^{-2}-2a^{4}z^{-2}+a^{2}z^{-2}$ (db)
Kauffman polynomial	$a^{7}z^{3}-a^{7}z+4a^{6}z^{4}-6a^{6}z^{2}+a^{6}z^{-2}-a^{6}+2a^{5}z^{7}-6a^{5}z^{5}+9a^{5}z^{3}-a^{5}z-2a^{5}z^{-1}+4a^{4}z^{8}-19a^{4}z^{6}+37a^{4}z^{4}-25a^{4}z^{2}+2a^{4}z^{-2}+2a^{4}+2a^{3}z^{9}-4a^{3}z^{7}-6a^{3}z^{5}+19a^{3}z^{3}-5a^{3}z-2a^{3}z^{-1}+8a^{2}z^{8}-35a^{2}z^{6}+z^{6}a^{-2}+51a^{2}z^{4}-3z^{4}a^{-2}-28a^{2}z^{2}+z^{2}a^{-2}+a^{2}z^{-2}+5a^{2}+2az^{9}-3az^{7}+3z^{7}a^{-1}-10az^{5}-10z^{5}a^{-1}+17az^{3}+6z^{3}a^{-1}-7az-2za^{-1}+4z^{8}-15z^{6}+15z^{4}-8z^{2}+3$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

2

-2

3

1

2

1

3

2

-1

1

6

3

2

-3

5

0

-5

4

0

-7

1

2

5

4

-9

3

0

-11

3

-13

1

-1

Integral Khovanov Homology

(db, data source)

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