L11n74

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-uv^{5}+uv^{4}-uv^{3}+uv^{2}-u-v^{7}+v^{5}-v^{4}+v^{3}-v^{2}}{{\sqrt {u}}v^{7/2}}}$ (db)
Jones polynomial	$3q^{9/2}-2q^{7/2}+q^{5/2}-q^{3/2}-{\frac {1}{q^{3/2}}}+q^{17/2}-q^{15/2}+2q^{13/2}-3q^{11/2}-{\sqrt {q}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+2za^{-7}+2z^{3}a^{-5}+5za^{-5}+3a^{-5}z^{-1}-z^{7}a^{-3}-8z^{5}a^{-3}-20z^{3}a^{-3}-19za^{-3}-7a^{-3}z^{-1}+z^{5}a^{-1}+6z^{3}a^{-1}+10za^{-1}+4a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}-3z^{2}a^{-10}+a^{-10}+z^{5}a^{-9}-2z^{3}a^{-9}+z^{6}a^{-8}-2z^{4}a^{-8}+z^{2}a^{-8}+z^{7}a^{-7}-4z^{5}a^{-7}+7z^{3}a^{-7}-2za^{-7}+z^{6}a^{-6}-2z^{4}a^{-6}+3z^{2}a^{-6}+z^{5}a^{-5}-3z^{3}a^{-5}+7za^{-5}-3a^{-5}z^{-1}+z^{8}a^{-4}-9z^{6}a^{-4}+26z^{4}a^{-4}-26z^{2}a^{-4}+7a^{-4}+z^{9}a^{-3}-10z^{7}a^{-3}+34z^{5}a^{-3}-49z^{3}a^{-3}+30za^{-3}-7a^{-3}z^{-1}+z^{8}a^{-2}-9z^{6}a^{-2}+25z^{4}a^{-2}-25z^{2}a^{-2}+7a^{-2}+z^{9}a^{-1}-9z^{7}a^{-1}+28z^{5}a^{-1}-37z^{3}a^{-1}+21za^{-1}-4a^{-1}z^{-1}$ (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

6

7

χ

18

1

-1

16

0

14

2

1

-1

12

1

10

2

0

8

1

-1

6

1

2

1

4

1

2

1

0

2

0

1

-2

1

-4

1

Integral Khovanov Homology

(db, data source)

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