L11a436

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(v-1)(w-1)\left(2uvw^{2}-uvw+2uv-uw^{2}-u+v^{2}w^{2}+v^{2}-2vw^{2}+vw-2v\right)}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{-11}+3q^{-10}-8q^{-9}+13q^{-8}-16q^{-7}+19q^{-6}-17q^{-5}+16q^{-4}-10q^{-3}+6q^{-2}-2q^{-1}+1$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{10}\left(-z^{2}\right)-a^{10}z^{-2}-2a^{10}+3a^{8}z^{4}+8a^{8}z^{2}+4a^{8}z^{-2}+8a^{8}-2a^{6}z^{6}-7a^{6}z^{4}-10a^{6}z^{2}-5a^{6}z^{-2}-10a^{6}-a^{4}z^{6}-2a^{4}z^{4}+2a^{4}z^{-2}+2a^{4}+a^{2}z^{4}+3a^{2}z^{2}+2a^{2}$ (db)
Kauffman polynomial	$z^{5}a^{13}-2z^{3}a^{13}+za^{13}+3z^{6}a^{12}-4z^{4}a^{12}+z^{2}a^{12}+6z^{7}a^{11}-10z^{5}a^{11}+8z^{3}a^{11}-5za^{11}+a^{11}z^{-1}+7z^{8}a^{10}-11z^{6}a^{10}+10z^{4}a^{10}-11z^{2}a^{10}-a^{10}z^{-2}+6a^{10}+4z^{9}a^{9}+5z^{7}a^{9}-29z^{5}a^{9}+41z^{3}a^{9}-25za^{9}+5a^{9}z^{-1}+z^{10}a^{8}+13z^{8}a^{8}-39z^{6}a^{8}+55z^{4}a^{8}-45z^{2}a^{8}-4a^{8}z^{-2}+21a^{8}+7z^{9}a^{7}-6z^{7}a^{7}-18z^{5}a^{7}+43z^{3}a^{7}-35za^{7}+9a^{7}z^{-1}+z^{10}a^{6}+9z^{8}a^{6}-32z^{6}a^{6}+49z^{4}a^{6}-44z^{2}a^{6}-5a^{6}z^{-2}+22a^{6}+3z^{9}a^{5}-3z^{7}a^{5}-5z^{5}a^{5}+14z^{3}a^{5}-15za^{5}+5a^{5}z^{-1}+3z^{8}a^{4}-6z^{6}a^{4}+4z^{4}a^{4}-6z^{2}a^{4}-2a^{4}z^{-2}+6a^{4}+2z^{7}a^{3}-5z^{5}a^{3}+2z^{3}a^{3}+za^{3}+z^{6}a^{2}-4z^{4}a^{2}+5z^{2}a^{2}-2a^{2}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

1

-1

1

-1

-3

5

1

4

-5

7

3

-4

-7

9

3

6

-9

8

7

-1

-11

11

9

2

-13

7

10

3

-15

6

9

-3

-17

2

7

5

-19

1

6

-5

-21

2

-23

1

-1

Integral Khovanov Homology

(db, data source)

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