L11a108

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {4uv^{4}-6uv^{3}+6uv^{2}-5uv+2u+2v^{5}-5v^{4}+6v^{3}-6v^{2}+4v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {2}{q^{25/2}}}+{\frac {5}{q^{23/2}}}-{\frac {8}{q^{21/2}}}+{\frac {12}{q^{19/2}}}-{\frac {15}{q^{17/2}}}+{\frac {14}{q^{15/2}}}-{\frac {14}{q^{13/2}}}+{\frac {10}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}(-z)-2a^{13}z^{-1}+3a^{11}z^{3}+8a^{11}z+4a^{11}z^{-1}-2a^{9}z^{5}-5a^{9}z^{3}-3a^{9}z-a^{9}z^{-1}-3a^{7}z^{5}-9a^{7}z^{3}-6a^{7}z-a^{7}z^{-1}-a^{5}z^{5}-2a^{5}z^{3}$ (db)
Kauffman polynomial	$a^{16}z^{6}-4a^{16}z^{4}+5a^{16}z^{2}-2a^{16}+2a^{15}z^{7}-6a^{15}z^{5}+4a^{15}z^{3}+a^{15}z+2a^{14}z^{8}-2a^{14}z^{6}-6a^{14}z^{4}+7a^{14}z^{2}-a^{14}+2a^{13}z^{9}-3a^{13}z^{7}+5a^{13}z^{5}-12a^{13}z^{3}+8a^{13}z-2a^{13}z^{-1}+a^{12}z^{10}+2a^{12}z^{8}-7a^{12}z^{6}+13a^{12}z^{4}-17a^{12}z^{2}+6a^{12}+6a^{11}z^{9}-18a^{11}z^{7}+35a^{11}z^{5}-36a^{11}z^{3}+16a^{11}z-4a^{11}z^{-1}+a^{10}z^{10}+6a^{10}z^{8}-20a^{10}z^{6}+30a^{10}z^{4}-19a^{10}z^{2}+5a^{10}+4a^{9}z^{9}-7a^{9}z^{7}+8a^{9}z^{5}-4a^{9}z^{3}+3a^{9}z-a^{9}z^{-1}+6a^{8}z^{8}-13a^{8}z^{6}+10a^{8}z^{4}-a^{8}+6a^{7}z^{7}-15a^{7}z^{5}+14a^{7}z^{3}-6a^{7}z+a^{7}z^{-1}+3a^{6}z^{6}-5a^{6}z^{4}+a^{5}z^{5}-2a^{5}z^{3}$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

3

1

-2

-8

4

-10

6

3

-3

-12

8

4

-14

7

0

-16

8

7

1

-18

4

7

3

-20

4

8

-4

-22

1

4

3

-24

1

4

-3

-26

1

-28

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$
$r=-11$	${\mathbb {Z} }$
$r=-10$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-9$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-7$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-6$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-5$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$	${\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$