L11a237

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_12,4,13,3 X_14,12,15,11 X_22,15,7,16 X_16,9,17,10 X_10,21,11,22 X_18,6,19,5 X_20,18,21,17 X₂₇₃₈ X_4,14,5,13 X_6,20,1,19
Gauss code	{1, -9, 2, -10, 7, -11}, {9, -1, 5, -6, 3, -2, 10, -3, 4, -5, 8, -7, 11, -8, 6, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{4}-5u^{2}v^{3}+8u^{2}v^{2}-5u^{2}v+u^{2}-2uv^{4}+9uv^{3}-15uv^{2}+9uv-2u+v^{4}-5v^{3}+8v^{2}-5v+1}{uv^{2}}}$ (db)
Jones polynomial	$-11q^{9/2}+{\frac {1}{q^{9/2}}}+17q^{7/2}-{\frac {3}{q^{7/2}}}-23q^{5/2}+{\frac {8}{q^{5/2}}}+25q^{3/2}-{\frac {15}{q^{3/2}}}-q^{13/2}+5q^{11/2}-25{\sqrt {q}}+{\frac {20}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-3z^{5}a^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+5az^{3}-5z^{3}a^{-1}+3z^{3}a^{-3}-z^{3}a^{-5}-2a^{3}z+6az-4za^{-1}+za^{-3}-a^{3}z^{-1}+3az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}-5az^{9}-13z^{9}a^{-1}-8z^{9}a^{-3}-5a^{2}z^{8}-21z^{8}a^{-2}-13z^{8}a^{-4}-13z^{8}-3a^{3}z^{7}+2az^{7}+12z^{7}a^{-1}-4z^{7}a^{-3}-11z^{7}a^{-5}-a^{4}z^{6}+8a^{2}z^{6}+46z^{6}a^{-2}+17z^{6}a^{-4}-5z^{6}a^{-6}+33z^{6}+7a^{3}z^{5}+12az^{5}+16z^{5}a^{-1}+27z^{5}a^{-3}+15z^{5}a^{-5}-z^{5}a^{-7}+3a^{4}z^{4}-a^{2}z^{4}-25z^{4}a^{-2}-4z^{4}a^{-4}+4z^{4}a^{-6}-21z^{4}-6a^{3}z^{3}-16az^{3}-21z^{3}a^{-1}-16z^{3}a^{-3}-5z^{3}a^{-5}-3a^{4}z^{2}-5a^{2}z^{2}+2z^{2}a^{-2}+3a^{3}z+10az+9za^{-1}+2za^{-3}+a^{4}+3a^{2}+3-a^{3}z^{-1}-3az^{-1}-2a^{-1}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

4

-4

10

7

1

6

8

10

4

-6

6

13

7

6

4

13

11

-2

2

12

0

9

14

5

-2

6

11

-5

-4

2

9

7

-6

1

6

-5

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{13}$
$r=3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$