L11a413

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_20,13,21,14 X_22,19,11,20 X_10,15,5,16 X_8,17,9,18 X_16,7,17,8 X_18,9,19,10 X_14,21,15,22 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 7, -6, 8, -5}, {11, -2, 3, -9, 5, -7, 6, -8, 4, -3, 9, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2uv^{2}w^{3}-2uv^{2}w^{2}+uv^{2}w+uvw^{4}-4uvw^{3}+6uvw^{2}-4uvw+uv+uw^{3}-3uw^{2}+3uw-u+v^{2}w^{4}-3v^{2}w^{3}+3v^{2}w^{2}-v^{2}w-vw^{4}+4vw^{3}-6vw^{2}+4vw-v-w^{3}+2w^{2}-2w}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{-14}+3q^{-13}-7q^{-12}+12q^{-11}-16q^{-10}+19q^{-9}-18q^{-8}+17q^{-7}-11q^{-6}+8q^{-5}-3q^{-4}+q^{-3}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}-a^{14}+4a^{12}z^{2}+4a^{12}z^{-2}+9a^{12}-6a^{10}z^{4}-20a^{10}z^{2}-5a^{10}z^{-2}-19a^{10}+3a^{8}z^{6}+13a^{8}z^{4}+19a^{8}z^{2}+2a^{8}z^{-2}+11a^{8}+a^{6}z^{6}+3a^{6}z^{4}+2a^{6}z^{2}$ (db)
Kauffman polynomial	$a^{17}z^{5}-2a^{17}z^{3}+a^{17}z+3a^{16}z^{6}-5a^{16}z^{4}+3a^{16}z^{2}-a^{16}+5a^{15}z^{7}-7a^{15}z^{5}+4a^{15}z^{3}-2a^{15}z+a^{15}z^{-1}+5a^{14}z^{8}-2a^{14}z^{6}-6a^{14}z^{4}+5a^{14}z^{2}-a^{14}z^{-2}+3a^{13}z^{9}+8a^{13}z^{7}-27a^{13}z^{5}+31a^{13}z^{3}-19a^{13}z+5a^{13}z^{-1}+a^{12}z^{10}+11a^{12}z^{8}-24a^{12}z^{6}+21a^{12}z^{4}-16a^{12}z^{2}-4a^{12}z^{-2}+13a^{12}+7a^{11}z^{9}-2a^{11}z^{7}-29a^{11}z^{5}+50a^{11}z^{3}-35a^{11}z+9a^{11}z^{-1}+a^{10}z^{10}+12a^{10}z^{8}-39a^{10}z^{6}+51a^{10}z^{4}-43a^{10}z^{2}-5a^{10}z^{-2}+22a^{10}+4a^{9}z^{9}-2a^{9}z^{7}-17a^{9}z^{5}+28a^{9}z^{3}-19a^{9}z+5a^{9}z^{-1}+6a^{8}z^{8}-19a^{8}z^{6}+26a^{8}z^{4}-23a^{8}z^{2}-2a^{8}z^{-2}+11a^{8}+3a^{7}z^{7}-7a^{7}z^{5}+3a^{7}z^{3}+a^{6}z^{6}-3a^{6}z^{4}+2a^{6}z^{2}$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-5

1

-7

3

1

-2

-9

5

-11

6

3

-3

-13

11

5

6

-15

8

7

-1

-17

11

10

1

-19

7

10

3

-21

5

9

-4

-23

2

7

5

-25

1

5

-4

-27

2

-29

1

-1

Integral Khovanov Homology

(db, data source)

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