L11n339

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_18,12,19,11 X_7,14,8,15 X_13,8,14,9 X_19,22,20,13 X_15,20,16,21 X_21,16,22,17 X_12,18,5,17 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, 3, -9}, {-5, 4, -7, 8, 9, -3, -6, 7, -8, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(2)-1)(t(3)-1)\left(t(3)t(2)^{2}+t(1)t(3)^{2}t(2)-t(3)^{2}t(2)+t(1)t(2)-t(2)-t(1)t(3)\right)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$1-2q^{-1}+4q^{-2}-3q^{-3}+4q^{-4}-2q^{-5}+3q^{-6}-q^{-8}+q^{-9}-q^{-10}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+2z^{2}a^{8}+4a^{8}z^{-2}+5a^{8}-2z^{2}a^{6}-5a^{6}z^{-2}-7a^{6}-z^{6}a^{4}-4z^{4}a^{4}-3z^{2}a^{4}+2a^{4}z^{-2}+a^{4}+z^{4}a^{2}+3z^{2}a^{2}+2a^{2}$ (db)
Kauffman polynomial	$z^{7}a^{11}-6z^{5}a^{11}+10z^{3}a^{11}-6za^{11}+a^{11}z^{-1}+z^{8}a^{10}-6z^{6}a^{10}+9z^{4}a^{10}-6z^{2}a^{10}-a^{10}z^{-2}+4a^{10}+2z^{7}a^{9}-16z^{5}a^{9}+33z^{3}a^{9}-22za^{9}+5a^{9}z^{-1}+z^{8}a^{8}-9z^{6}a^{8}+22z^{4}a^{8}-22z^{2}a^{8}-4a^{8}z^{-2}+15a^{8}+2z^{7}a^{7}-15z^{5}a^{7}+34z^{3}a^{7}-30za^{7}+9a^{7}z^{-1}+z^{8}a^{6}-5z^{6}a^{6}+11z^{4}a^{6}-19z^{2}a^{6}-5a^{6}z^{-2}+16a^{6}+3z^{7}a^{5}-12z^{5}a^{5}+15z^{3}a^{5}-13za^{5}+5a^{5}z^{-1}+z^{8}a^{4}-z^{6}a^{4}-6z^{4}a^{4}+2z^{2}a^{4}-2a^{4}z^{-2}+4a^{4}+2z^{7}a^{3}-7z^{5}a^{3}+4z^{3}a^{3}+za^{3}+z^{6}a^{2}-4z^{4}a^{2}+5z^{2}a^{2}-2a^{2}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

1

-1

1

-1

-3

3

1

2

-5

2

3

1

-7

1

3

1

-9

1

2

-11

1

5

3

1

-13

1

2

3

-15

1

2

-1

-17

1

0

-19

0

-21

1

-1

Integral Khovanov Homology

(db, data source)

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