L11n347

L11n346

L11n348

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n347 at Knotilus!

Link Presentations

[edit Notes on L11n347's Link Presentations]

Planar diagram presentation	X6172 X3,15,4,14 X11,20,12,21 X7,18,8,19 X17,22,18,13 X16,9,17,10 X10,15,11,16 X19,12,20,5 X21,8,22,9 X2536 X13,1,14,4
Gauss code	{1, -10, -2, 11}, {10, -1, -4, 9, 6, -7, -3, 8}, {-11, 2, 7, -6, -5, 4, -8, 3, -9, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{2}w^{3}-uv^{2}w^{2}+uv^{2}w-uv^{2}-uvw+uv+uw^{2}+v^{3}(-w)-v^{2}w^{3}+v^{2}w^{2}+vw^{3}-vw^{2}+vw-v}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$1-2q^{-1}+4q^{-2}-3q^{-3}+4q^{-4}-3q^{-5}+3q^{-6}-q^{-7}+q^{-9}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{8}z^{2}+a^{8}z^{-2}+2a^{8}-2a^{6}z^{2}-2a^{6}z^{-2}-5a^{6}-a^{4}z^{6}-4a^{4}z^{4}-3a^{4}z^{2}+a^{4}z^{-2}+a^{4}+a^{2}z^{4}+3a^{2}z^{2}+2a^{2}$ (db)
Kauffman polynomial	$a^{10}z^{6}-6a^{10}z^{4}+8a^{10}z^{2}-2a^{10}-a^{9}z^{5}+2a^{9}z^{3}+a^{9}z-a^{8}z^{6}+4a^{8}z^{4}-3a^{8}z^{2}-a^{8}z^{-2}+3a^{8}+a^{7}z^{7}-5a^{7}z^{5}+10a^{7}z^{3}-8a^{7}z+2a^{7}z^{-1}+a^{6}z^{8}-4a^{6}z^{6}+8a^{6}z^{4}-13a^{6}z^{2}-2a^{6}z^{-2}+9a^{6}+3a^{5}z^{7}-11a^{5}z^{5}+12a^{5}z^{3}-8a^{5}z+2a^{5}z^{-1}+a^{4}z^{8}-a^{4}z^{6}-6a^{4}z^{4}+3a^{4}z^{2}-a^{4}z^{-2}+3a^{4}+2a^{3}z^{7}-7a^{3}z^{5}+4a^{3}z^{3}+a^{3}z+a^{2}z^{6}-4a^{2}z^{4}+5a^{2}z^{2}-2a^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                     1 1 -1                   1   -1 -3                 3 1   2 -5               2 3     1 -7           1 3 1       1 -9           1 2         1 -11       1 4 3           0 -13         2             2 -15     1 2               -1 -17 1                     1 -19 1                     1

Integral Khovanov Homology (db, data source)

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

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.

L11n346 L11n348