L11n455

L11n454

L11n456

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n455 at Knotilus!

Link Presentations

[edit Notes on L11n455's Link Presentations]

Planar diagram presentation	X6172 X14,5,15,6 X12,4,13,3 X2,9,3,10 X7,19,8,18 X17,9,18,8 X10,13,5,14 X19,22,20,17 X21,11,22,16 X11,21,12,20 X4,16,1,15
Gauss code	{1, -4, 3, -11}, {2, -1, -5, 6, 4, -7}, {-10, -3, 7, -2, 11, 9}, {-6, 5, -8, 10, -9, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)t(3)^{2}t(2)^{2}+t(3)^{2}t(2)^{2}-t(1)t(2)^{2}+2t(1)t(3)t(2)^{2}+t(1)t(4)t(2)^{2}-t(1)t(3)t(4)t(2)^{2}-2t(3)^{2}t(2)+t(1)t(2)-t(1)t(3)t(2)+t(3)t(2)+t(3)^{2}t(4)t(2)-2t(1)t(4)t(2)+t(1)t(3)t(4)t(2)-t(3)t(4)t(2)+t(3)^{2}-t(3)-t(3)^{2}t(4)+t(1)t(4)+2t(3)t(4)-t(4)}{{\sqrt {t(1)}}t(2)t(3){\sqrt {t(4)}}}}$ (db)
Jones polynomial	$2q^{9/2}-6q^{7/2}+{\frac {1}{q^{7/2}}}+5q^{5/2}-{\frac {4}{q^{5/2}}}-9q^{3/2}+{\frac {5}{q^{3/2}}}-q^{11/2}+7{\sqrt {q}}-{\frac {8}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-2z^{5}a^{-1}+3az^{3}-8z^{3}a^{-1}+3z^{3}a^{-3}-a^{3}z+6az-12za^{-1}+8za^{-3}-za^{-5}+3az^{-1}-8a^{-1}z^{-1}+7a^{-3}z^{-1}-2a^{-5}z^{-1}+az^{-3}-3a^{-1}z^{-3}+3a^{-3}z^{-3}-a^{-5}z^{-3}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-6z^{8}a^{-2}-2z^{8}a^{-4}-4z^{8}-5az^{7}-7z^{7}a^{-1}-3z^{7}a^{-3}-z^{7}a^{-5}-2a^{2}z^{6}+19z^{6}a^{-2}+7z^{6}a^{-4}+10z^{6}+17az^{5}+39z^{5}a^{-1}+27z^{5}a^{-3}+5z^{5}a^{-5}+2a^{2}z^{4}-6z^{4}a^{-2}-3z^{4}a^{-4}-z^{4}-4a^{3}z^{3}-24az^{3}-52z^{3}a^{-1}-42z^{3}a^{-3}-10z^{3}a^{-5}-a^{4}z^{2}-2a^{2}z^{2}-18z^{2}a^{-2}-9z^{2}a^{-4}-10z^{2}+2a^{3}z+16az+31za^{-1}+27za^{-3}+10za^{-5}+19a^{-2}+10a^{-4}+10-5az^{-1}-12a^{-1}z^{-1}-12a^{-3}z^{-1}-5a^{-5}z^{-1}-6a^{-2}z^{-2}-3a^{-4}z^{-2}-3z^{-2}+az^{-3}+3a^{-1}z^{-3}+3a^{-3}z^{-3}+a^{-5}z^{-3}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 χ 12                   1 1 10                 1   -1 8               5 1   4 6             2 3     1 4           7 3       4 2       1 3 4         2 0       6 5           1 -2   1 2 4             3 -4   3 4               -1 -6   3                 3 -8 1                   -1

Integral Khovanov Homology (db, data source)

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

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

L11n454 L11n456