L11n389

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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_5,12,6,13 X₃₈₄₉ X_13,22,14,19 X_9,20,10,21 X_19,10,20,11 X_21,14,22,15 X_11,18,12,5 X_15,2,16,3
Gauss code	{1, 11, -5, -3}, {-8, 7, -9, 6}, {-4, -1, 2, 5, -7, 8, -10, 4, -6, 9, -11, -2, 3, 10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(3)-1)\left(t(1)t(3)^{4}-t(1)t(3)^{3}+2t(1)t(2)t(3)^{3}+t(1)t(3)^{2}-t(1)t(2)t(3)^{2}+t(2)t(3)^{2}-t(3)^{2}-t(2)t(3)+2t(3)+t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$q^{-3}-2q^{-4}+5q^{-5}-6q^{-6}+9q^{-7}-7q^{-8}+8q^{-9}-6q^{-10}+3q^{-11}-q^{-12}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}+z^{2}a^{12}+4a^{12}z^{-2}+6a^{12}-3z^{4}a^{10}-13z^{2}a^{10}-5a^{10}z^{-2}-15a^{10}+2z^{6}a^{8}+9z^{4}a^{8}+12z^{2}a^{8}+2a^{8}z^{-2}+8a^{8}+z^{6}a^{6}+4z^{4}a^{6}+4z^{2}a^{6}+a^{6}$ (db)
Kauffman polynomial	$z^{3}a^{15}-2za^{15}+a^{15}z^{-1}+3z^{4}a^{14}-3z^{2}a^{14}-a^{14}z^{-2}+2a^{14}+2z^{7}a^{13}-7z^{5}a^{13}+18z^{3}a^{13}-15za^{13}+5a^{13}z^{-1}+3z^{8}a^{12}-12z^{6}a^{12}+26z^{4}a^{12}-25z^{2}a^{12}-4a^{12}z^{-2}+14a^{12}+z^{9}a^{11}+3z^{7}a^{11}-22z^{5}a^{11}+44z^{3}a^{11}-33za^{11}+9a^{11}z^{-1}+6z^{8}a^{10}-24z^{6}a^{10}+42z^{4}a^{10}-42z^{2}a^{10}-5a^{10}z^{-2}+21a^{10}+z^{9}a^{9}+3z^{7}a^{9}-21z^{5}a^{9}+30z^{3}a^{9}-20za^{9}+5a^{9}z^{-1}+3z^{8}a^{8}-11z^{6}a^{8}+15z^{4}a^{8}-16z^{2}a^{8}-2a^{8}z^{-2}+9a^{8}+2z^{7}a^{7}-6z^{5}a^{7}+3z^{3}a^{7}+z^{6}a^{6}-4z^{4}a^{6}+4z^{2}a^{6}-a^{6}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-5

1

-7

2

1

-1

-9

3

-11

3

2

-1

-13

6

3

-15

3

5

2

-17

5

4

1

-19

1

3

2

-21

2

5

-3

-23

2

-25

1

-1

Integral Khovanov Homology

(db, data source)

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