L11n442

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_11,19,12,18 X_3,11,4,10 X_9,1,10,4 X_7,17,8,16 X_15,5,16,8 X_13,20,14,21 X_19,15,20,22 X_21,12,22,13 X_17,9,18,14
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 10, -8, 11}, {-7, 6, -11, 3, -9, 8, -10, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uvw^{2}x-uvw^{2}+uvwx^{2}-4uvwx+2uvw-uvx^{2}+2uvx-uv-uwx^{2}+2uwx+ux^{2}-ux-vw^{2}x+vw^{2}+2vwx-vw+w^{2}\left(-x^{2}\right)+2w^{2}x-w^{2}+2wx^{2}-4wx+w-x^{2}+x}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$-10q^{9/2}+10q^{7/2}-14q^{5/2}+10q^{3/2}-{\frac {3}{q^{3/2}}}+q^{15/2}-3q^{13/2}+6q^{11/2}-10{\sqrt {q}}+{\frac {5}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$2z^{5}a^{-3}-5z^{3}a^{-1}+7z^{3}a^{-3}-3z^{3}a^{-5}+3az-11za^{-1}+13za^{-3}-6za^{-5}+za^{-7}+3az^{-1}-9a^{-1}z^{-1}+10a^{-3}z^{-1}-5a^{-5}z^{-1}+a^{-7}z^{-1}+az^{-3}-3a^{-1}z^{-3}+3a^{-3}z^{-3}-a^{-5}z^{-3}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-3z^{4}a^{-8}+3z^{2}a^{-8}-a^{-8}+3z^{7}a^{-7}-9z^{5}a^{-7}+9z^{3}a^{-7}-6za^{-7}+2a^{-7}z^{-1}+3z^{8}a^{-6}-3z^{6}a^{-6}-11z^{4}a^{-6}+14z^{2}a^{-6}-6a^{-6}+z^{9}a^{-5}+10z^{7}a^{-5}-40z^{5}a^{-5}+47z^{3}a^{-5}-a^{-5}z^{-3}-30za^{-5}+11a^{-5}z^{-1}+8z^{8}a^{-4}-14z^{6}a^{-4}-9z^{4}a^{-4}+28z^{2}a^{-4}+3a^{-4}z^{-2}-18a^{-4}+z^{9}a^{-3}+14z^{7}a^{-3}-53z^{5}a^{-3}+74z^{3}a^{-3}-3a^{-3}z^{-3}-49za^{-3}+18a^{-3}z^{-1}+5z^{8}a^{-2}-7z^{6}a^{-2}-4z^{4}a^{-2}+24z^{2}a^{-2}+6a^{-2}z^{-2}-21a^{-2}+7z^{7}a^{-1}-22z^{5}a^{-1}+6az^{3}+42z^{3}a^{-1}-az^{-3}-3a^{-1}z^{-3}-10az-35za^{-1}+5az^{-1}+14a^{-1}z^{-1}+3z^{6}-3z^{4}+7z^{2}+3z^{-2}-9$ (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		\

-2

-1

0

1

2

3

4

5

6

7

χ

16

1

-1

14

2

12

4

1

-3

10

6

2

4

8

5

0

6

9

5

4

8

4

2

6

0

2

7

5

-2

1

3

-2

-4

3

Integral Khovanov Homology

(db, data source)

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