L11a464

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_16,6,17,5 X₈₄₉₃ X_20,9,21,10 X_10,19,11,20 X_22,18,15,17 X_18,22,19,21 X_14,16,5,15 X_2,12,3,11
Gauss code	{1, -11, 5, -3}, {10, -4, 8, -9, 7, -6, 9, -8}, {4, -1, 2, -5, 6, -7, 11, -2, 3, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2(u-1)(v-1)(w-1)^{3}}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$q^{6}-5q^{5}+10q^{4}-14q^{3}+19q^{2}-20q+21-16q^{-1}+12q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}-z^{6}+2a^{2}z^{4}-z^{4}a^{-2}+z^{4}a^{-4}-2z^{4}-a^{4}z^{2}+4a^{2}z^{2}+2z^{2}a^{-2}-5z^{2}-a^{4}+4a^{2}+4a^{-2}-a^{-4}-6+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$z^{6}a^{-6}-z^{4}a^{-6}+5z^{7}a^{-5}+a^{5}z^{5}-10z^{5}a^{-5}-2a^{5}z^{3}+2z^{3}a^{-5}+a^{5}z+za^{-5}+9z^{8}a^{-4}+3a^{4}z^{6}-22z^{6}a^{-4}-6a^{4}z^{4}+11z^{4}a^{-4}+5a^{4}z^{2}+z^{2}a^{-4}-2a^{4}-2a^{-4}+7z^{9}a^{-3}+4a^{3}z^{7}-11z^{7}a^{-3}-3a^{3}z^{5}-2z^{5}a^{-3}-3a^{3}z^{3}+z^{3}a^{-3}+3a^{3}z+3za^{-3}+2z^{10}a^{-2}+4a^{2}z^{8}+11z^{8}a^{-2}+3a^{2}z^{6}-33z^{6}a^{-2}-17a^{2}z^{4}+16z^{4}a^{-2}+18a^{2}z^{2}+10z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-8a^{2}-8a^{-2}+4az^{9}+11z^{9}a^{-1}-20z^{7}a^{-1}-3az^{5}+9z^{5}a^{-1}-az^{3}-z^{3}a^{-1}+4az+4za^{-1}-2az^{-1}-2a^{-1}z^{-1}+2z^{10}+6z^{8}-10z^{6}-7z^{4}+22z^{2}+2z^{-2}-11$ (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

13

1

11

4

-4

9

6

1

5

7

8

4

-4

5

11

6

5

3

9

8

-1

1

12

11

1

-1

8

13

5

-3

4

8

-4

-5

2

8

6

-7

1

4

-3

-9

2

-11

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$