L11a25

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{5}-4uv^{4}+6uv^{3}-7uv^{2}+6uv-3u-3v^{5}+6v^{4}-7v^{3}+6v^{2}-4v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$16q^{9/2}-17q^{7/2}+17q^{5/2}-{\frac {1}{q^{5/2}}}-15q^{3/2}+{\frac {2}{q^{3/2}}}+q^{17/2}-3q^{15/2}+7q^{13/2}-12q^{11/2}+10{\sqrt {q}}-{\frac {7}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+2za^{-7}-2z^{5}a^{-5}-5z^{3}a^{-5}-2za^{-5}+a^{-5}z^{-1}+z^{7}a^{-3}+3z^{5}a^{-3}+2z^{3}a^{-3}-a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}+3az-5za^{-1}+2az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-3z^{9}a^{-1}-8z^{9}a^{-3}-5z^{9}a^{-5}-8z^{8}a^{-2}-15z^{8}a^{-4}-9z^{8}a^{-6}-2z^{8}-az^{7}+8z^{7}a^{-1}+16z^{7}a^{-3}-2z^{7}a^{-5}-9z^{7}a^{-7}+35z^{6}a^{-2}+49z^{6}a^{-4}+14z^{6}a^{-6}-6z^{6}a^{-8}+6z^{6}+5az^{5}-4z^{5}a^{-1}-z^{5}a^{-3}+25z^{5}a^{-5}+14z^{5}a^{-7}-3z^{5}a^{-9}-38z^{4}a^{-2}-49z^{4}a^{-4}-7z^{4}a^{-6}+6z^{4}a^{-8}-z^{4}a^{-10}-3z^{4}-9az^{3}-2z^{3}a^{-1}-z^{3}a^{-3}-22z^{3}a^{-5}-12z^{3}a^{-7}+2z^{3}a^{-9}+14z^{2}a^{-2}+23z^{2}a^{-4}-4z^{2}a^{-8}+z^{2}a^{-10}-4z^{2}+7az+3za^{-1}-4za^{-3}+3za^{-5}+3za^{-7}-3a^{-4}+a^{-8}+3-2az^{-1}-2a^{-1}z^{-1}+a^{-3}z^{-1}+a^{-5}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

18

1

-1

16

2

14

5

1

-4

12

7

2

5

10

9

5

-4

8

7

1

6

9

0

4

6

8

-2

2

5

10

5

0

2

5

-3

-2

5

-4

1

2

-1

-6

1

Integral Khovanov Homology

(db, data source)

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