L11n125

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_9,22,10,5 X_7,19,8,18 X_17,9,18,8 X_19,13,20,12 X_11,21,12,20 X_15,10,16,11 X_21,16,22,17 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, -3, 8, -7, 6, 11, -2, -8, 9, -5, 4, -6, 7, -9, 3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{5}-t(1)t(2)^{4}-3t(2)^{4}+4t(2)^{3}+4t(1)t(2)^{2}-3t(1)t(2)-t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}+{\frac {5}{q^{7/2}}}-q^{5/2}-{\frac {5}{q^{5/2}}}+q^{3/2}+{\frac {4}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {4}{q^{11/2}}}-{\frac {2}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}+2z^{3}a^{5}+3za^{5}+2a^{5}z^{-1}-z^{5}a^{3}-4z^{3}a^{3}-8za^{3}-4a^{3}z^{-1}+z^{5}a+6z^{3}a+7za+3az^{-1}-z^{3}a^{-1}-3za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{5}z^{9}-a^{3}z^{9}-3a^{6}z^{8}-4a^{4}z^{8}-2a^{2}z^{8}-z^{8}-3a^{7}z^{7}-a^{5}z^{7}+2a^{3}z^{7}-az^{7}-z^{7}a^{-1}-a^{8}z^{6}+10a^{6}z^{6}+18a^{4}z^{6}+14a^{2}z^{6}+7z^{6}+11a^{7}z^{5}+18a^{5}z^{5}+12a^{3}z^{5}+11az^{5}+6z^{5}a^{-1}+3a^{8}z^{4}-5a^{6}z^{4}-23a^{4}z^{4}-27a^{2}z^{4}-12z^{4}-9a^{7}z^{3}-27a^{5}z^{3}-34a^{3}z^{3}-25az^{3}-9z^{3}a^{-1}-a^{8}z^{2}+a^{6}z^{2}+10a^{4}z^{2}+15a^{2}z^{2}+7z^{2}+3a^{7}z+15a^{5}z+23a^{3}z+16az+5za^{-1}-a^{6}-2a^{4}-3a^{2}-1-2a^{5}z^{-1}-4a^{3}z^{-1}-3az^{-1}-a^{-1}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

6

1

4

0

2

1

-1

0

3

1

2

-2

3

2

1

-2

-4

3

1

-6

3

0

-8

3

1

-10

2

4

2

-12

1

2

-1

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

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