L11n375

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,10,6,11 X₃₈₄₉ X_17,22,18,19 X_11,20,12,21 X_19,12,20,13 X_21,18,22,5 X_9,16,10,17 X_13,2,14,3
Gauss code	{1, 11, -5, -3}, {-8, 7, -9, 6}, {-4, -1, 2, 5, -10, 4, -7, 8, -11, -2, 3, 10, -6, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(3)-1)\left(t(1)t(3)^{4}-t(1)t(3)^{3}+2t(1)t(2)t(3)^{3}-t(2)t(3)+2t(3)+t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$q^{-3}-2q^{-4}+4q^{-5}-3q^{-6}+6q^{-7}-5q^{-8}+5q^{-9}-3q^{-10}+2q^{-11}-q^{-12}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}+a^{12}z^{2}+4a^{12}z^{-2}+6a^{12}-3a^{10}z^{4}-14a^{10}z^{2}-5a^{10}z^{-2}-15a^{10}+2a^{8}z^{6}+10a^{8}z^{4}+14a^{8}z^{2}+2a^{8}z^{-2}+8a^{8}+a^{6}z^{6}+4a^{6}z^{4}+3a^{6}z^{2}+a^{6}$ (db)
Kauffman polynomial	$z^{3}a^{15}-2za^{15}+a^{15}z^{-1}+2z^{4}a^{14}-4z^{2}a^{14}-a^{14}z^{-2}+2a^{14}+z^{7}a^{13}-5z^{5}a^{13}+14z^{3}a^{13}-15za^{13}+5a^{13}z^{-1}+2z^{8}a^{12}-11z^{6}a^{12}+25z^{4}a^{12}-25z^{2}a^{12}-4a^{12}z^{-2}+14a^{12}+z^{9}a^{11}-z^{7}a^{11}-14z^{5}a^{11}+40z^{3}a^{11}-33za^{11}+9a^{11}z^{-1}+5z^{8}a^{10}-26z^{6}a^{10}+47z^{4}a^{10}-40z^{2}a^{10}-5a^{10}z^{-2}+21a^{10}+z^{9}a^{9}-16z^{5}a^{9}+30z^{3}a^{9}-20za^{9}+5a^{9}z^{-1}+3z^{8}a^{8}-14z^{6}a^{8}+20z^{4}a^{8}-16z^{2}a^{8}-2a^{8}z^{-2}+9a^{8}+2z^{7}a^{7}-7z^{5}a^{7}+3z^{3}a^{7}+z^{6}a^{6}-4z^{4}a^{6}+3z^{2}a^{6}-a^{6}$ (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

2

-11

1

2

1

-13

1

6

2

3

-15

2

3

1

-17

4

0

-19

1

2

-21

2

3

-1

-23

1

-25

1

-1

Integral Khovanov Homology

(db, data source)

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