L11a401

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)^{2}(w-1)^{2}\left(w^{2}-w+1\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{-8}+5q^{-7}-12q^{-6}+20q^{-5}-27q^{-4}+q^{3}+32q^{-3}-5q^{2}-30q^{-2}+12q+28q^{-1}-19$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{6}z^{4}-a^{6}z^{2}+2a^{4}z^{6}+5a^{4}z^{4}+3a^{4}z^{2}+a^{4}z^{-2}-a^{2}z^{8}-4a^{2}z^{6}-6a^{2}z^{4}-3a^{2}z^{2}-2a^{2}z^{-2}-a^{2}+z^{6}+2z^{4}+z^{2}+z^{-2}+1$ (db)
Kauffman polynomial	$4a^{4}z^{10}+4a^{2}z^{10}+13a^{5}z^{9}+24a^{3}z^{9}+11az^{9}+17a^{6}z^{8}+28a^{4}z^{8}+22a^{2}z^{8}+11z^{8}+12a^{7}z^{7}-6a^{5}z^{7}-37a^{3}z^{7}-14az^{7}+5z^{7}a^{-1}+5a^{8}z^{6}-25a^{6}z^{6}-73a^{4}z^{6}-67a^{2}z^{6}+z^{6}a^{-2}-23z^{6}+a^{9}z^{5}-14a^{7}z^{5}-17a^{5}z^{5}+2a^{3}z^{5}-4az^{5}-8z^{5}a^{-1}-3a^{8}z^{4}+14a^{6}z^{4}+51a^{4}z^{4}+51a^{2}z^{4}-z^{4}a^{-2}+16z^{4}+5a^{7}z^{3}+10a^{5}z^{3}+8a^{3}z^{3}+6az^{3}+3z^{3}a^{-1}-4a^{6}z^{2}-12a^{4}z^{2}-12a^{2}z^{2}-4z^{2}-a^{3}z-az+a^{4}+a^{2}+1+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

4

-4

3

8

1

7

1

11

4

-7

-1

17

8

9

-3

15

13

-2

-5

17

15

2

-7

12

17

5

-9

8

15

-7

-11

4

12

8

-13

1

8

-7

-15

4

-17

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-3$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=-2$	${\mathbb {Z} }^{17}\oplus {\mathbb {Z} }_{2}^{15}$	${\mathbb {Z} }^{17}$
$r=-1$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{15}$	${\mathbb {Z} }^{15}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{15}$	${\mathbb {Z} }^{17}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$