L11a57

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a57 at Knotilus!
	[edit L11a57 Quick Notes]

[edit L11a57 Further Notes and Views]

Link Presentations

[edit Notes on L11a57's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}-3t(2)+1\right)\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-4q^{9/2}+{\frac {3}{q^{9/2}}}+8q^{7/2}-{\frac {7}{q^{7/2}}}-13q^{5/2}+{\frac {12}{q^{5/2}}}+17q^{3/2}-{\frac {16}{q^{3/2}}}+q^{11/2}-{\frac {1}{q^{11/2}}}-20{\sqrt {q}}+{\frac {18}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+2z^{3}a^{-3}-6a^{3}z+za^{-3}-3a^{3}z^{-1}-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+9az^{3}-7z^{3}a^{-1}+9az-5za^{-1}+4az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-3a^{3}z^{9}-8az^{9}-5z^{9}a^{-1}-3a^{4}z^{8}-11a^{2}z^{8}-10z^{8}a^{-2}-18z^{8}-a^{5}z^{7}+3a^{3}z^{7}+5az^{7}-10z^{7}a^{-1}-11z^{7}a^{-3}+11a^{4}z^{6}+44a^{2}z^{6}+10z^{6}a^{-2}-8z^{6}a^{-4}+51z^{6}+4a^{5}z^{5}+17a^{3}z^{5}+42az^{5}+47z^{5}a^{-1}+14z^{5}a^{-3}-4z^{5}a^{-5}-14a^{4}z^{4}-47a^{2}z^{4}+3z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}-38z^{4}-6a^{5}z^{3}-32a^{3}z^{3}-60az^{3}-44z^{3}a^{-1}-8z^{3}a^{-3}+2z^{3}a^{-5}+7a^{4}z^{2}+19a^{2}z^{2}-2z^{2}a^{-2}-2z^{2}a^{-4}+12z^{2}+4a^{5}z+18a^{3}z+26az+15za^{-1}+3za^{-3}-a^{4}-3a^{2}-a^{-2}-2-a^{5}z^{-1}-3a^{3}z^{-1}-4az^{-1}-2a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

3

8

5

1

-4

6

8

3

5

4

9

5

-4

2

11

8

3

0

9

11

2

-2

7

9

-2

-4

5

9

4

-6

2

7

-5

-8

1

5

4

-10

2

-2

-12

1

Integral Khovanov Homology

(db, data source)

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