L11n279

L11n278

L11n280

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n279 at Knotilus!

Link Presentations

[edit Notes on L11n279's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X18,10,19,9 X17,11,18,22 X11,21,12,20 X21,17,22,16 X15,1,16,4 X10,20,5,19
Gauss code	{1, -4, -3, 10}, {-2, -1, 5, 3, 6, -11}, {-8, 2, 4, -5, -10, 9, -7, -6, 11, 8, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2uvw^{2}-2uvw+uv-uw^{2}+uw+v^{2}(-w)+v^{2}-vw^{2}+2vw-2v}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-2q^{6}+3q^{5}-4q^{4}+6q^{3}-4q^{2}+5q-2+2q^{-1}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-2z^{4}a^{-2}-7z^{2}a^{-2}+4z^{2}a^{-4}+2z^{2}-10a^{-2}+8a^{-4}-2a^{-6}+4-5a^{-2}z^{-2}+4a^{-4}z^{-2}-a^{-6}z^{-2}+2z^{-2}$ (db)
Kauffman polynomial	$3z^{3}a^{-7}-5za^{-7}+a^{-7}z^{-1}+z^{6}a^{-6}-2z^{2}a^{-6}-a^{-6}z^{-2}+3a^{-6}+2z^{7}a^{-5}-8z^{5}a^{-5}+20z^{3}a^{-5}-18za^{-5}+5a^{-5}z^{-1}+z^{8}a^{-4}-3z^{6}a^{-4}+10z^{4}a^{-4}-14z^{2}a^{-4}-4a^{-4}z^{-2}+12a^{-4}+3z^{7}a^{-3}-11z^{5}a^{-3}+24z^{3}a^{-3}-24za^{-3}+9a^{-3}z^{-1}+z^{8}a^{-2}-4z^{6}a^{-2}+13z^{4}a^{-2}-21z^{2}a^{-2}-5a^{-2}z^{-2}+15a^{-2}+z^{7}a^{-1}-3z^{5}a^{-1}+7z^{3}a^{-1}-11za^{-1}+5a^{-1}z^{-1}+3z^{4}-9z^{2}-2z^{-2}+7$ (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   \ -2 -1 0 1 2 3 4 5 χ 13               2 -2 11             1   1 9           3 2   -1 7         3 1     2 5       2 4       2 3     3 2         1 1   1 4           3 -1 1 1             0 -3 2               2

Integral Khovanov Homology (db, data source)

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

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.

L11n278 L11n280