L11n354

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,18,12,19 X_7,16,8,17 X_15,8,16,9 X_13,21,14,20 X_19,22,20,15 X_21,13,22,12 X_17,14,18,5 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, -9, 3, -7, 6, -8, 7}, {10, -1, -4, 5, 11, -2, -3, 8, -6, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(3)-1)\left(t(2)t(3)^{3}-t(3)^{3}+t(1)t(2)^{2}t(3)^{2}-t(1)t(2)t(3)^{2}-t(2)t(3)^{2}-t(1)t(2)t(3)-t(2)t(3)+t(3)-t(1)t(2)^{2}+t(1)t(2)\right)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{-10}+2q^{-9}-2q^{-8}+3q^{-7}-q^{-6}+2q^{-5}+q^{-2}-q^{-1}+1$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{10}+2a^{8}z^{2}+a^{8}z^{-2}+2a^{8}+a^{6}z^{2}-2a^{6}z^{-2}-2a^{6}-a^{4}z^{6}-5a^{4}z^{4}-5a^{4}z^{2}+a^{4}z^{-2}-a^{4}+a^{2}z^{4}+4a^{2}z^{2}+2a^{2}$ (db)
Kauffman polynomial	$a^{11}z^{7}-5a^{11}z^{5}+6a^{11}z^{3}-2a^{11}z+2a^{10}z^{8}-11a^{10}z^{6}+17a^{10}z^{4}-11a^{10}z^{2}+3a^{10}+a^{9}z^{9}-3a^{9}z^{7}-6a^{9}z^{5}+16a^{9}z^{3}-7a^{9}z+4a^{8}z^{8}-25a^{8}z^{6}+46a^{8}z^{4}-35a^{8}z^{2}-a^{8}z^{-2}+12a^{8}+a^{7}z^{9}-4a^{7}z^{7}-4a^{7}z^{5}+19a^{7}z^{3}-14a^{7}z+2a^{7}z^{-1}+2a^{6}z^{8}-14a^{6}z^{6}+28a^{6}z^{4}-26a^{6}z^{2}-2a^{6}z^{-2}+12a^{6}+a^{5}z^{7}-8a^{5}z^{5}+14a^{5}z^{3}-10a^{5}z+2a^{5}z^{-1}+a^{4}z^{6}-6a^{4}z^{4}+4a^{4}z^{2}-a^{4}z^{-2}+2a^{4}+a^{3}z^{7}-5a^{3}z^{5}+5a^{3}z^{3}-a^{3}z+a^{2}z^{6}-5a^{2}z^{4}+6a^{2}z^{2}-2a^{2}$ (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

0

-3

1

2

1

0

-5

1

-7

2

3

1

0

-9

2

-11

2

4

1

-13

2

1

2

-15

1

3

1

-17

1

0

-19

1

-21

1

-1

Integral Khovanov Homology

(db, data source)

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