L11n407

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_22,12,19,11 X_10,4,11,3 X_5,21,6,20 X_21,5,22,18 X_12,20,13,19 X_14,9,15,10 X_2,14,3,13 X_8,15,9,16
Gauss code	{1, -10, 5, -3}, {8, 6, -7, -4}, {-6, -1, 2, -11, 9, -5, 4, -8, 10, -9, 11, -2, 3, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2(t(1)-1)(t(3)-1)^{2}(t(2)t(3)+1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$3q^{5}-5q^{4}+9q^{3}-9q^{2}+11q-10+8q^{-1}-5q^{-2}+3q^{-3}-q^{-4}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$a^{-6}z^{-2}+a^{-6}-z^{4}a^{-4}-3z^{2}a^{-4}-2a^{-4}z^{-2}-4a^{-4}+z^{6}a^{-2}-a^{2}z^{4}+3z^{4}a^{-2}-2a^{2}z^{2}+3z^{2}a^{-2}+a^{-2}z^{-2}+3a^{-2}+z^{6}+3z^{4}+2z^{2}$ (db)
Kauffman polynomial	$2az^{9}+2z^{9}a^{-1}+3a^{2}z^{8}+6z^{8}a^{-2}+9z^{8}+a^{3}z^{7}-3az^{7}+4z^{7}a^{-1}+8z^{7}a^{-3}-13a^{2}z^{6}-12z^{6}a^{-2}+7z^{6}a^{-4}-32z^{6}-4a^{3}z^{5}-10az^{5}-25z^{5}a^{-1}-16z^{5}a^{-3}+3z^{5}a^{-5}+17a^{2}z^{4}+4z^{4}a^{-2}-12z^{4}a^{-4}+33z^{4}+5a^{3}z^{3}+18az^{3}+20z^{3}a^{-1}+7z^{3}a^{-3}-6a^{2}z^{2}-3z^{2}a^{-2}+11z^{2}a^{-4}+6z^{2}a^{-6}-14z^{2}-2a^{3}z-6az-6za^{-1}+2za^{-3}+4za^{-5}+a^{2}-a^{-2}-6a^{-4}-4a^{-6}+3-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-2}z^{-2}+2a^{-4}z^{-2}+a^{-6}z^{-2}$ (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

11

3

9

4

2

-2

7

5

1

4

5

4

0

3

7

5

2

1

5

6

1

-1

3

5

-2

-3

2

5

3

-5

1

3

-2

-7

2

-9

1

-1

Integral Khovanov Homology

(db, data source)

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