L11n398

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(w-1)\left(uvw^{3}+uvw^{2}-uvw+uv-uw^{2}+uw+vw^{3}-vw^{2}+w^{4}-w^{3}+w^{2}+w\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$q^{-3}-q^{-5}+4q^{-6}-4q^{-7}+6q^{-8}-5q^{-9}+6q^{-10}-4q^{-11}+2q^{-12}-q^{-13}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}+4a^{12}z^{-2}+4a^{12}-5z^{2}a^{10}-5a^{10}z^{-2}-9a^{10}+z^{4}a^{8}+z^{2}a^{8}+2a^{8}z^{-2}+2a^{8}+z^{6}a^{6}+6z^{4}a^{6}+8z^{2}a^{6}+3a^{6}$ (db)
Kauffman polynomial	$z^{7}a^{15}-5z^{5}a^{15}+8z^{3}a^{15}-5za^{15}+a^{15}z^{-1}+2z^{8}a^{14}-9z^{6}a^{14}+12z^{4}a^{14}-8z^{2}a^{14}-a^{14}z^{-2}+4a^{14}+z^{9}a^{13}+z^{7}a^{13}-21z^{5}a^{13}+36z^{3}a^{13}-23za^{13}+5a^{13}z^{-1}+6z^{8}a^{12}-28z^{6}a^{12}+41z^{4}a^{12}-34z^{2}a^{12}-4a^{12}z^{-2}+18a^{12}+z^{9}a^{11}+3z^{7}a^{11}-32z^{5}a^{11}+57z^{3}a^{11}-39za^{11}+9a^{11}z^{-1}+4z^{8}a^{10}-21z^{6}a^{10}+38z^{4}a^{10}-37z^{2}a^{10}-5a^{10}z^{-2}+21a^{10}+3z^{7}a^{9}-17z^{5}a^{9}+30z^{3}a^{9}-20za^{9}+5a^{9}z^{-1}-z^{6}a^{8}+3z^{4}a^{8}-3z^{2}a^{8}-2a^{8}z^{-2}+5a^{8}-z^{5}a^{7}+z^{3}a^{7}+za^{7}+z^{6}a^{6}-6z^{4}a^{6}+8z^{2}a^{6}-3a^{6}$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-5

1

-7

1

-9

2

1

-1

-11

3

-13

4

5

1

0

-15

3

1

2

-17

2

4

1

-19

4

3

1

-21

1

3

2

-23

1

3

-2

-25

1

-27

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-7$	$i=-5$	$i=-3$
$r=-11$		${\mathbb {Z} }$
$r=-10$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-9$		${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-8$		${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{4}$
$r=-7$		${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-6$		${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-5$	${\mathbb {Z} }$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$