L11n59

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_14,8,15,7 X_11,19,12,18 X_19,5,20,22 X_15,21,16,20 X_21,17,22,16 X_17,13,18,12 X_8,14,9,13 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, -4, 8, 9, -3, -6, 7, -8, 4, -5, 6, -7, 5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{5}-t(1)t(2)^{4}+t(2)^{4}+4t(1)t(2)^{3}-4t(2)^{3}-4t(1)t(2)^{2}+4t(2)^{2}+t(1)t(2)-t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-6q^{9/2}+7q^{7/2}-6q^{5/2}-{\frac {1}{q^{5/2}}}+5q^{3/2}+q^{15/2}-3q^{13/2}+5q^{11/2}-4{\sqrt {q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-1}+z^{5}a^{-3}+az^{3}-7z^{3}a^{-1}+3z^{3}a^{-3}-2z^{3}a^{-5}+4az-10za^{-1}+5za^{-3}-2za^{-5}+za^{-7}+3az^{-1}-5a^{-1}z^{-1}+2a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-3z^{4}a^{-8}+2z^{2}a^{-8}+3z^{7}a^{-7}-10z^{5}a^{-7}+7z^{3}a^{-7}-za^{-7}+3z^{8}a^{-6}-9z^{6}a^{-6}+5z^{4}a^{-6}-3z^{2}a^{-6}+a^{-6}+z^{9}a^{-5}+z^{7}a^{-5}-11z^{5}a^{-5}+8z^{3}a^{-5}-za^{-5}+4z^{8}a^{-4}-15z^{6}a^{-4}+19z^{4}a^{-4}-10z^{2}a^{-4}+z^{9}a^{-3}-3z^{7}a^{-3}+2z^{5}a^{-3}+3z^{3}a^{-3}-3za^{-3}+2a^{-3}z^{-1}+z^{8}a^{-2}-5z^{6}a^{-2}+7z^{4}a^{-2}+5z^{2}a^{-2}-5a^{-2}+az^{7}-7az^{5}-4z^{5}a^{-1}+14az^{3}+16z^{3}a^{-1}-11az-14za^{-1}+3az^{-1}+5a^{-1}z^{-1}-4z^{4}+10z^{2}-5$ (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

16

1

-1

14

2

12

3

1

-2

10

3

2

1

8

4

3

-1

6

1

3

-1

4

3

4

1

2

1

3

-1

0

4

-2

1

0

-4

1

-6

1

Integral Khovanov Homology

(db, data source)

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