L11a532

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Link Presentations

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uvwx^{3}-uvwx^{2}+uvwx-uvw-uvx^{3}+2uvx^{2}-2uvx+2uv-2uwx^{3}+2uwx^{2}-2uwx+uw+2ux^{3}-3ux^{2}+3ux-2u-2vwx^{3}+3vwx^{2}-3vwx+2vw+vx^{3}-2vx^{2}+2vx-2v+2wx^{3}-2wx^{2}+2wx-w-x^{3}+x^{2}-x+1}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+10q^{7/2}-14q^{5/2}+17q^{3/2}-19{\sqrt {q}}+{\frac {16}{\sqrt {q}}}-{\frac {16}{q^{3/2}}}+{\frac {7}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {1}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+12az^{3}-9z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-11a^{3}z+22az-15za^{-1}+3za^{-3}+3a^{5}z^{-1}-14a^{3}z^{-1}+22az^{-1}-14a^{-1}z^{-1}+3a^{-3}z^{-1}+2a^{5}z^{-3}-7a^{3}z^{-3}+9az^{-3}-5a^{-1}z^{-3}+a^{-3}z^{-3}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-a^{3}z^{9}-6az^{9}-5z^{9}a^{-1}-a^{4}z^{8}-a^{2}z^{8}-11z^{8}a^{-2}-11z^{8}-a^{5}z^{7}-3a^{3}z^{7}+6az^{7}-6z^{7}a^{-1}-14z^{7}a^{-3}+a^{4}z^{6}-2a^{2}z^{6}+9z^{6}a^{-2}-10z^{6}a^{-4}+16z^{6}+6a^{5}z^{5}+23a^{3}z^{5}+21az^{5}+28z^{5}a^{-1}+20z^{5}a^{-3}-4z^{5}a^{-5}+10a^{4}z^{4}+36a^{2}z^{4}+19z^{4}a^{-2}+10z^{4}a^{-4}-z^{4}a^{-6}+34z^{4}-14a^{5}z^{3}-42a^{3}z^{3}-42az^{3}-22z^{3}a^{-1}-8z^{3}a^{-3}-26a^{4}z^{2}-73a^{2}z^{2}-34z^{2}a^{-2}-6z^{2}a^{-4}-75z^{2}+16a^{5}z+39a^{3}z+37az+16za^{-1}+2za^{-3}+23a^{4}+60a^{2}+24a^{-2}+4a^{-4}+58-9a^{5}z^{-1}-23a^{3}z^{-1}-24az^{-1}-12a^{-1}z^{-1}-2a^{-3}z^{-1}-7a^{4}z^{-2}-19a^{2}z^{-2}-7a^{-2}z^{-2}-a^{-4}z^{-2}-18z^{-2}+2a^{5}z^{-3}+7a^{3}z^{-3}+9az^{-3}+5a^{-1}z^{-3}+a^{-3}z^{-3}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

3

8

7

1

-6

6

7

3

4

10

7

-3

2

9

7

2

0

11

14

3

-2

5

0

-4

2

11

9

-6

4

5

-1

-8

1

6

5

-10

0

-12

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{6}$	${\mathbb {Z} }^{4}$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$