L11n337

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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_7,14,8,15 X_13,8,14,9 X_19,22,20,13 X_15,20,16,21 X_21,16,22,17 X_17,12,18,5 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 $u$ , $v$ , $w$ , ...)	${\frac {(t(2)-1)\left(t(1)t(3)^{3}+t(1)t(2)t(3)^{3}+2t(2)^{2}t(3)^{2}-2t(1)t(3)^{2}-2t(2)^{2}t(3)+2t(1)t(3)+t(2)^{2}+t(2)\right)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-13}+2q^{-12}-4q^{-11}+6q^{-10}-5q^{-9}+6q^{-8}-4q^{-7}+4q^{-6}-q^{-5}+q^{-3}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}+4a^{12}z^{-2}+4a^{12}-5a^{10}z^{2}-5a^{10}z^{-2}-9a^{10}+a^{8}z^{4}+a^{8}z^{2}+2a^{8}z^{-2}+2a^{8}+a^{6}z^{6}+6a^{6}z^{4}+8a^{6}z^{2}+3a^{6}$ (db)
Kauffman polynomial	$a^{15}z^{7}-5a^{15}z^{5}+8a^{15}z^{3}-5a^{15}z+a^{15}z^{-1}+2a^{14}z^{8}-9a^{14}z^{6}+12a^{14}z^{4}-8a^{14}z^{2}-a^{14}z^{-2}+4a^{14}+a^{13}z^{9}+a^{13}z^{7}-21a^{13}z^{5}+36a^{13}z^{3}-23a^{13}z+5a^{13}z^{-1}+6a^{12}z^{8}-28a^{12}z^{6}+41a^{12}z^{4}-34a^{12}z^{2}-4a^{12}z^{-2}+18a^{12}+a^{11}z^{9}+3a^{11}z^{7}-32a^{11}z^{5}+57a^{11}z^{3}-39a^{11}z+9a^{11}z^{-1}+4a^{10}z^{8}-21a^{10}z^{6}+38a^{10}z^{4}-37a^{10}z^{2}-5a^{10}z^{-2}+21a^{10}+3a^{9}z^{7}-17a^{9}z^{5}+30a^{9}z^{3}-20a^{9}z+5a^{9}z^{-1}-a^{8}z^{6}+3a^{8}z^{4}-3a^{8}z^{2}-2a^{8}z^{-2}+5a^{8}-a^{7}z^{5}+a^{7}z^{3}+a^{7}z+a^{6}z^{6}-6a^{6}z^{4}+8a^{6}z^{2}-3a^{6}$ (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)

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