L11n457

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n457 at Knotilus!
	[edit L11n457 Quick Notes]

[edit L11n457 Further Notes and Views]

Link Presentations

[edit Notes on L11n457's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_15,18,16,11 X_20,9,21,10 X_22,13,19,14 X_14,21,15,22 X_10,19,5,20 X_17,8,18,9 X_7,16,8,17 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {7, -4, 6, -5}, {10, -1, -9, 8, 4, -7}, {11, -2, 5, -6, -3, 9, -8, 3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(4)^{2}t(3)^{2}+t(2)t(4)^{2}t(3)^{2}-t(4)^{2}t(3)^{2}-t(1)t(4)t(3)^{2}+t(1)t(2)t(4)t(3)^{2}-t(2)t(4)t(3)^{2}+t(4)t(3)^{2}-t(1)t(4)^{2}t(3)+t(1)t(2)t(4)^{2}t(3)-t(2)t(4)^{2}t(3)+t(4)^{2}t(3)-t(1)t(3)+t(1)t(2)t(3)-t(2)t(3)+2t(1)t(4)t(3)-3t(1)t(2)t(4)t(3)+2t(2)t(4)t(3)-3t(4)t(3)+t(3)+t(1)-t(1)t(2)+t(2)-t(1)t(4)+t(1)t(2)t(4)-t(2)t(4)+t(4)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {12}{q^{13/2}}}+{\frac {10}{q^{15/2}}}-{\frac {11}{q^{17/2}}}+{\frac {6}{q^{19/2}}}-{\frac {5}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}z^{-3}-2za^{11}-5a^{11}z^{-1}-3a^{11}z^{-3}+5z^{3}a^{9}+14za^{9}+10a^{9}z^{-1}+3a^{9}z^{-3}-3z^{5}a^{7}-11z^{3}a^{7}-12za^{7}-5a^{7}z^{-1}-a^{7}z^{-3}-z^{5}a^{5}-2z^{3}a^{5}$ (db)
Kauffman polynomial	$-z^{4}a^{14}-5z^{5}a^{13}+5z^{3}a^{13}+4za^{13}-5a^{13}z^{-1}+a^{13}z^{-3}-z^{8}a^{12}-2z^{6}a^{12}+3z^{4}a^{12}-6z^{2}a^{12}-3a^{12}z^{-2}+10a^{12}-z^{9}a^{11}-z^{7}a^{11}-z^{5}a^{11}-3z^{3}a^{11}+15za^{11}-12a^{11}z^{-1}+3a^{11}z^{-3}-5z^{8}a^{10}+6z^{6}a^{10}+4z^{4}a^{10}-20z^{2}a^{10}-6a^{10}z^{-2}+19a^{10}-z^{9}a^{9}-7z^{7}a^{9}+22z^{5}a^{9}-29z^{3}a^{9}+23za^{9}-12a^{9}z^{-1}+3a^{9}z^{-3}-4z^{8}a^{8}+5z^{6}a^{8}+5z^{4}a^{8}-14z^{2}a^{8}-3a^{8}z^{-2}+10a^{8}-6z^{7}a^{7}+17z^{5}a^{7}-19z^{3}a^{7}+12za^{7}-5a^{7}z^{-1}+a^{7}z^{-3}-3z^{6}a^{6}+5z^{4}a^{6}-z^{5}a^{5}+2z^{3}a^{5}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

3

1

-2

-8

4

-10

4

3

-1

-12

8

4

-14

4

6

2

-16

7

6

1

-18

3

8

5

-20

2

3

-1

-22

4

-24

1

-1

Integral Khovanov Homology

(db, data source)

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