L11a481

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(w-1)\left(2uvw^{2}-4uvw+2uv-uw^{2}+3uw-4u-4vw^{2}+3vw-v+2w^{2}-4w+2\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+5q^{7}-10q^{6}+15q^{5}-19q^{4}+21q^{3}-20q^{2}+17q-10+7q^{-1}-2q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+z^{4}a^{-2}+z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}-z^{2}a^{-4}-4z^{2}+2a^{2}+a^{-2}-a^{-4}+a^{-6}-3+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$z^{5}a^{-9}+5z^{6}a^{-8}-5z^{4}a^{-8}+10z^{7}a^{-7}-15z^{5}a^{-7}+4z^{3}a^{-7}+10z^{8}a^{-6}-11z^{6}a^{-6}-4z^{4}a^{-6}+6z^{2}a^{-6}-2a^{-6}+5z^{9}a^{-5}+8z^{7}a^{-5}-28z^{5}a^{-5}+18z^{3}a^{-5}-4za^{-5}+z^{10}a^{-4}+15z^{8}a^{-4}-24z^{6}a^{-4}-z^{4}a^{-4}+12z^{2}a^{-4}-2a^{-4}+7z^{9}a^{-3}+2z^{7}a^{-3}-30z^{5}a^{-3}+34z^{3}a^{-3}-12za^{-3}+z^{10}a^{-2}+8z^{8}a^{-2}+a^{2}z^{6}-13z^{6}a^{-2}-4a^{2}z^{4}+6a^{2}z^{2}+6z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-4a^{2}-2a^{-2}+2z^{9}a^{-1}+2az^{7}+6z^{7}a^{-1}-4az^{5}-22z^{5}a^{-1}+20z^{3}a^{-1}+4az-4za^{-1}-2az^{-1}-2a^{-1}z^{-1}+3z^{8}-4z^{6}-2z^{4}+6z^{2}+2z^{-2}-5$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

17

1

-1

15

4

13

6

1

-5

11

9

4

5

9

11

7

-4

7

10

8

2

5

10

11

1

3

7

10

-3

1

5

12

7

-1

2

5

-3

5

-5

1

2

-1

-7

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-4$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{7}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$