L11a139

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_22,10,7,9 X₂₇₃₈ X_4,15,5,16 X_12,5,13,6 X_16,12,17,11 X_6,18,1,17 X_14,20,15,19 X_20,14,21,13 X_18,21,19,22
Gauss code	{1, -4, 2, -5, 6, -8}, {4, -1, 3, -2, 7, -6, 10, -9, 5, -7, 8, -11, 9, -10, 11, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{4}-5u^{2}v^{3}+8u^{2}v^{2}-5u^{2}v+u^{2}-2uv^{4}+10uv^{3}-15uv^{2}+10uv-2u+v^{4}-5v^{3}+8v^{2}-5v+1}{uv^{2}}}$ (db)
Jones polynomial	$-q^{13/2}+4q^{11/2}-9q^{9/2}+16q^{7/2}-22q^{5/2}+25q^{3/2}-26{\sqrt {q}}+{\frac {22}{\sqrt {q}}}-{\frac {17}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-3z^{5}a^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+3az^{3}-5z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-3za^{-1}+3za^{-3}-za^{-5}+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-5z^{4}a^{-6}+2z^{2}a^{-6}+8z^{7}a^{-5}-11z^{5}a^{-5}+7z^{3}a^{-5}-2za^{-5}+10z^{8}a^{-4}+a^{4}z^{6}-12z^{6}a^{-4}-a^{4}z^{4}+5z^{4}a^{-4}+7z^{9}a^{-3}+5a^{3}z^{7}+4z^{7}a^{-3}-10a^{3}z^{5}-25z^{5}a^{-3}+5a^{3}z^{3}+22z^{3}a^{-3}+a^{3}z-6za^{-3}-a^{3}z^{-1}+2z^{10}a^{-2}+9a^{2}z^{8}+21z^{8}a^{-2}-19a^{2}z^{6}-49z^{6}a^{-2}+11a^{2}z^{4}+34z^{4}a^{-2}-2a^{2}z^{2}-8z^{2}a^{-2}+a^{2}+7az^{9}+14z^{9}a^{-1}-2az^{7}-11z^{7}a^{-1}-22az^{5}-25z^{5}a^{-1}+17az^{3}+26z^{3}a^{-1}-az-6za^{-1}-az^{-1}+2z^{10}+20z^{8}-53z^{6}+36z^{4}-8z^{2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

6

1

5

8

10

3

-7

6

12

6

4

13

10

-3

2

13

12

1

0

10

14

4

-2

7

12

-5

-4

4

11

7

-6

1

6

-5

-8

4

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$