L11n244

L11n243

L11n245

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n244 at Knotilus!

Link Presentations

[edit Notes on L11n244's Link Presentations]

Planar diagram presentation	X12,1,13,2 X10,11,1,12 X5,15,6,14 X9,19,10,18 X17,3,18,2 X7,16,8,17 X3849 X15,20,16,21 X22,13,11,14 X19,4,20,5 X21,7,22,6
Gauss code	{1, 5, -7, 10, -3, 11, -6, 7, -4, -2}, {2, -1, 9, 3, -8, 6, -5, 4, -10, 8, -11, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$0$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}+{\frac {1}{q^{7/2}}}-q^{5/2}-{\frac {1}{q^{5/2}}}+2q^{3/2}-{\frac {1}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}-2{\sqrt {q}}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z^{3}a^{5}+2za^{5}-z^{5}a^{3}-5z^{3}a^{3}-6za^{3}+z^{5}a+5z^{3}a+6za+az^{-1}-z^{3}a^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-a^{4}z^{8}-3a^{2}z^{8}-2z^{8}-a^{5}z^{7}+4a^{3}z^{7}+4az^{7}-z^{7}a^{-1}-2a^{6}z^{6}+5a^{4}z^{6}+18a^{2}z^{6}+11z^{6}-a^{7}z^{5}+5a^{5}z^{5}+5a^{3}z^{5}+4az^{5}+5z^{5}a^{-1}+8a^{6}z^{4}-2a^{4}z^{4}-24a^{2}z^{4}-14z^{4}+3a^{7}z^{3}-5a^{5}z^{3}-21a^{3}z^{3}-19az^{3}-6z^{3}a^{-1}-4a^{6}z^{2}-4a^{4}z^{2}+4a^{2}z^{2}+4z^{2}+4a^{5}z+12a^{3}z+12az+4za^{-1}+1-az^{-1}-a^{-1}z^{-1}$ (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 χ 6                     1 1 4                   1   -1 2                 1 1   0 0             3 2 1     2 -2           1 3 1       1 -4         2 2 2         2 -6       1 2 1           0 -8     1 2 1             0 -10   1 1 1               -1 -12   1                   -1 -14 1                     1

Integral Khovanov Homology (db, data source)

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

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.

L11n243 L11n245