L11a125

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(2v^{4}-v^{3}+2v^{2}-v+2\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{23/2}-3q^{21/2}+5q^{19/2}-7q^{17/2}+10q^{15/2}-10q^{13/2}+9q^{11/2}-8q^{9/2}+5q^{7/2}-4q^{5/2}+q^{3/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$z^{5}a^{-9}+3z^{3}a^{-9}+za^{-9}-z^{7}a^{-7}-4z^{5}a^{-7}-3z^{3}a^{-7}+2za^{-7}+2a^{-7}z^{-1}-z^{7}a^{-5}-5z^{5}a^{-5}-9z^{3}a^{-5}-10za^{-5}-5a^{-5}z^{-1}+z^{5}a^{-3}+5z^{3}a^{-3}+7za^{-3}+3a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-6}-z^{10}a^{-8}-z^{9}a^{-5}-4z^{9}a^{-7}-3z^{9}a^{-9}-z^{8}a^{-4}+3z^{8}a^{-6}-4z^{8}a^{-10}-z^{7}a^{-3}+z^{7}a^{-5}+15z^{7}a^{-7}+9z^{7}a^{-9}-4z^{7}a^{-11}+3z^{6}a^{-4}-5z^{6}a^{-6}+5z^{6}a^{-8}+9z^{6}a^{-10}-4z^{6}a^{-12}+6z^{5}a^{-3}+10z^{5}a^{-5}-18z^{5}a^{-7}-14z^{5}a^{-9}+5z^{5}a^{-11}-3z^{5}a^{-13}+2z^{4}a^{-4}+13z^{4}a^{-6}-5z^{4}a^{-8}-10z^{4}a^{-10}+5z^{4}a^{-12}-z^{4}a^{-14}-12z^{3}a^{-3}-20z^{3}a^{-5}+7z^{3}a^{-7}+9z^{3}a^{-9}-2z^{3}a^{-11}+4z^{3}a^{-13}-9z^{2}a^{-4}-13z^{2}a^{-6}+z^{2}a^{-8}+3z^{2}a^{-10}-z^{2}a^{-12}+z^{2}a^{-14}+10za^{-3}+15za^{-5}+3za^{-7}-2za^{-9}+5a^{-4}+5a^{-6}-a^{-10}-3a^{-3}z^{-1}-5a^{-5}z^{-1}-2a^{-7}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

24

1

-1

22

2

20

3

1

-2

18

4

2

16

6

3

-3

14

4

0

12

5

6

1

10

3

4

-1

8

2

5

3

6

2

3

-1

4

1

4

3

2

0

1

Integral Khovanov Homology

(db, data source)

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