L11n278

L11n277

L11n279

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n278 at Knotilus!

Link Presentations

[edit Notes on L11n278's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X18,10,19,9 X17,11,18,22 X11,21,12,20 X21,17,22,16 X4,15,1,16 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 $u$, $v$, $w$, ...)	${\frac {uv^{2}w^{2}-uv^{2}w+uv-vw^{2}+w-1}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{4}+q^{3}-q^{2}+q+q^{-1}+q^{-2}+q^{-3}+q^{-4}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z^{6}-a^{2}z^{4}-z^{4}a^{-2}+6z^{4}-6a^{2}z^{2}-4z^{2}a^{-2}+11z^{2}+2a^{4}-9a^{2}-3a^{-2}+10+2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}+4z^{-2}$ (db)
Kauffman polynomial	$z^{8}a^{-2}+z^{8}+az^{7}+2z^{7}a^{-1}+z^{7}a^{-3}-a^{2}z^{6}-6z^{6}a^{-2}-7z^{6}-a^{3}z^{5}-8az^{5}-13z^{5}a^{-1}-6z^{5}a^{-3}+a^{4}z^{4}+7a^{2}z^{4}+10z^{4}a^{-2}+16z^{4}+6a^{3}z^{3}+21az^{3}+25z^{3}a^{-1}+10z^{3}a^{-3}-5a^{4}z^{2}-17a^{2}z^{2}-6z^{2}a^{-2}-18z^{2}-11a^{3}z-24az-18za^{-1}-5za^{-3}+7a^{4}+15a^{2}+3a^{-2}+12+5a^{3}z^{-1}+9az^{-1}+5a^{-1}z^{-1}+a^{-3}z^{-1}-2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-4z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 9                   1 -1 7                     0 5               1 1   0 3           1 1       0 1             1       1 -1       1 3 1         1 -3     1   1           2 -5     3 1             2 -7 1   1               2 -9 1                   1

Integral Khovanov Homology (db, data source)

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

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.

L11n277 L11n279