L11a207

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_14,9,15,10 X₆₇₁₈ X_22,15,7,16 X_16,6,17,5 X_4,22,5,21 X_10,4,11,3 X_20,18,21,17 X_12,20,13,19 X_18,12,19,11 X_2,14,3,13
Gauss code	{1, -11, 7, -6, 5, -3}, {3, -1, 2, -7, 10, -9, 11, -2, 4, -5, 8, -10, 9, -8, 6, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {\left(uv^{2}-2uv+u-v^{2}+2v-2\right)\left(2uv^{2}-2uv+u-v^{2}+2v-1\right)}{uv^{2}}}$ (db)
Jones polynomial	$q^{15/2}-5q^{13/2}+11q^{11/2}-17q^{9/2}+23q^{7/2}-27q^{5/2}+25q^{3/2}-23{\sqrt {q}}+{\frac {16}{\sqrt {q}}}-{\frac {9}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-5}+z^{3}a^{-5}-za^{-5}-z^{7}a^{-3}-2z^{5}a^{-3}+z^{3}a^{-3}+3za^{-3}-a^{-3}z^{-1}-z^{7}a^{-1}+az^{5}-3z^{5}a^{-1}+2az^{3}-4z^{3}a^{-1}+az-2za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-z^{4}a^{-8}+5z^{7}a^{-7}-9z^{5}a^{-7}+3z^{3}a^{-7}+10z^{8}a^{-6}-22z^{6}a^{-6}+12z^{4}a^{-6}+9z^{9}a^{-5}-12z^{7}a^{-5}-5z^{5}a^{-5}+7z^{3}a^{-5}-2za^{-5}+3z^{10}a^{-4}+16z^{8}a^{-4}-47z^{6}a^{-4}+29z^{4}a^{-4}-2z^{2}a^{-4}+17z^{9}a^{-3}-26z^{7}a^{-3}+a^{3}z^{5}-a^{3}z^{3}+12z^{3}a^{-3}-5za^{-3}-a^{-3}z^{-1}+3z^{10}a^{-2}+16z^{8}a^{-2}+4a^{2}z^{6}-39z^{6}a^{-2}-5a^{2}z^{4}+23z^{4}a^{-2}+2a^{2}z^{2}-2z^{2}a^{-2}+a^{-2}+8z^{9}a^{-1}+8az^{7}-z^{7}a^{-1}-11az^{5}-16z^{5}a^{-1}+7az^{3}+16z^{3}a^{-1}-2az-5za^{-1}-a^{-1}z^{-1}+10z^{8}-11z^{6}+2z^{4}+2z^{2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

16

1

-1

14

4

12

7

1

-6

10

4

6

8

13

7

-6

6

14

10

4

12

14

2

11

13

-2

0

6

13

7

-2

3

10

-7

-4

1

6

5

-6

3

-3

-8

1

Integral Khovanov Homology

(db, data source)

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