L11n417

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_15,21,16,20 X_5,15,6,14 X_13,5,14,4 X_19,7,20,12 X_11,19,12,18 X_17,13,18,22 X_21,17,22,16 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -10, 2, 5, -4, -11}, {10, -1, 11, -2, -7, 6}, {-5, 4, -3, 9, -8, 7, -6, 3, -9, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}vw^{3}-u^{2}w^{3}+u^{2}w^{2}-u^{2}w+u^{2}+uv^{2}w^{3}-uvw^{3}+uv-u-v^{2}w^{3}+v^{2}w^{2}-v^{2}w+v^{2}-v}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{7}+2q^{6}-3q^{5}+4q^{4}-4q^{3}+5q^{2}-3q+4-q^{-1}+q^{-2}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}-z^{6}a^{-4}-5z^{4}a^{-2}-4z^{4}a^{-4}+z^{4}a^{-6}+z^{4}-9z^{2}a^{-2}-2z^{2}a^{-4}+4z^{2}a^{-6}+4z^{2}-10a^{-2}+4a^{-4}+2a^{-6}-a^{-8}+5-5a^{-2}z^{-2}+4a^{-4}z^{-2}-a^{-6}z^{-2}+2z^{-2}$ (db)
Kauffman polynomial	$za^{-9}+2z^{2}a^{-8}-a^{-8}+z^{5}a^{-7}-z^{3}a^{-7}-2za^{-7}+a^{-7}z^{-1}+3z^{6}a^{-6}-10z^{4}a^{-6}+6z^{2}a^{-6}-a^{-6}z^{-2}+5z^{7}a^{-5}-23z^{5}a^{-5}+31z^{3}a^{-5}-19za^{-5}+5a^{-5}z^{-1}+4z^{8}a^{-4}-19z^{6}a^{-4}+27z^{4}a^{-4}-21z^{2}a^{-4}-4a^{-4}z^{-2}+13a^{-4}+z^{9}a^{-3}+z^{7}a^{-3}-25z^{5}a^{-3}+50z^{3}a^{-3}-35za^{-3}+9a^{-3}z^{-1}+5z^{8}a^{-2}-29z^{6}a^{-2}+55z^{4}a^{-2}-46z^{2}a^{-2}-5a^{-2}z^{-2}+22a^{-2}+z^{9}a^{-1}-4z^{7}a^{-1}-z^{5}a^{-1}+18z^{3}a^{-1}-19za^{-1}+5a^{-1}z^{-1}+z^{8}-7z^{6}+18z^{4}-21z^{2}-2z^{-2}+11$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

15

1

-1

13

1

11

3

2

-1

9

2

1

7

3

0

5

2

1

3

2

4

2

1

2

1

-1

1

4

3

-3

0

-5

1

Integral Khovanov Homology

(db, data source)

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