L11n261

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_7,15,8,14 X_13,5,14,8 X_18,12,19,11 X_22,20,9,19 X_20,16,21,15 X_16,22,17,21 X_12,18,13,17 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, 5, -9, -4, 3, 7, -8, 9, -5, 6, -7, 8, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uvw^{3}-uvw+uv-uw^{3}+2uw^{2}-2uw+u-vw^{3}+2vw^{2}-2vw+v-w^{3}+w^{2}-1}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{10}+q^{9}-q^{8}-q^{7}+3q^{6}-3q^{5}+5q^{4}-3q^{3}+5q^{2}-2q+1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-a^{-10}z^{-2}-a^{-10}+2z^{2}a^{-8}+3a^{-8}z^{-2}+5a^{-8}-2z^{2}a^{-6}-2a^{-6}z^{-2}-5a^{-6}-z^{6}a^{-4}-4z^{4}a^{-4}-4z^{2}a^{-4}-a^{-4}z^{-2}-2a^{-4}+z^{4}a^{-2}+3z^{2}a^{-2}+a^{-2}z^{-2}+3a^{-2}$ (db)
Kauffman polynomial	$z^{8}a^{-4}+z^{8}a^{-6}+z^{8}a^{-8}+z^{8}a^{-10}+2z^{7}a^{-3}+4z^{7}a^{-5}+3z^{7}a^{-7}+2z^{7}a^{-9}+z^{7}a^{-11}+z^{6}a^{-2}+z^{6}a^{-4}-2z^{6}a^{-6}-8z^{6}a^{-8}-6z^{6}a^{-10}-6z^{5}a^{-3}-15z^{5}a^{-5}-20z^{5}a^{-7}-17z^{5}a^{-9}-6z^{5}a^{-11}-4z^{4}a^{-2}-14z^{4}a^{-4}-3z^{4}a^{-6}+15z^{4}a^{-8}+8z^{4}a^{-10}+z^{3}a^{-3}+16z^{3}a^{-5}+42z^{3}a^{-7}+37z^{3}a^{-9}+10z^{3}a^{-11}+6z^{2}a^{-2}+12z^{2}a^{-4}+2z^{2}a^{-6}-7z^{2}a^{-8}-3z^{2}a^{-10}+4za^{-3}-10za^{-5}-34za^{-7}-27za^{-9}-7za^{-11}-4a^{-2}-3a^{-4}+4a^{-6}+5a^{-8}+a^{-10}-2a^{-3}z^{-1}+2a^{-5}z^{-1}+10a^{-7}z^{-1}+8a^{-9}z^{-1}+2a^{-11}z^{-1}+a^{-2}z^{-2}+a^{-4}z^{-2}-2a^{-6}z^{-2}-3a^{-8}z^{-2}-a^{-10}z^{-2}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

21

1

-1

19

0

17

1

0

15

3

1

-2

13

1

2

11

4

5

1

0

9

2

1

2

7

1

4

1

2

5

4

2

3

1

4

3

1

-1

1

Integral Khovanov Homology

(db, data source)

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