L11n309

L11n308

L11n310

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n309 at Knotilus!

Link Presentations

[edit Notes on L11n309's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X7,17,8,16 X9,21,10,20 X15,9,16,8 X19,5,20,10 X18,13,19,14 X22,17,11,18 X21,15,22,14 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 5, -4, 6}, {11, -2, 7, 9, -5, 3, 8, -7, -6, 4, -9, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{2}w-uv^{2}+uvw^{2}-4uvw+3uv-uw^{2}+3uw-u+v^{2}w^{2}-3v^{2}w+v^{2}-3vw^{2}+4vw-v+w^{2}-w}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{3}+5q^{2}-6q+10-10q^{-1}+10q^{-2}-8q^{-3}+6q^{-4}-3q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}-3z^{2}a^{4}-2a^{4}+2z^{4}a^{2}+3z^{2}a^{2}+a^{2}z^{-2}+3a^{2}+z^{4}-2z^{2}-2z^{-2}-4-z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+3a^{6}z^{2}-a^{6}+3a^{5}z^{7}-9a^{5}z^{5}+7a^{5}z^{3}-2a^{5}z+3a^{4}z^{8}-5a^{4}z^{6}-5a^{4}z^{4}+6a^{4}z^{2}-a^{4}+a^{3}z^{9}+6a^{3}z^{7}-24a^{3}z^{5}+20a^{3}z^{3}+z^{3}a^{-3}-6a^{3}z+6a^{2}z^{8}-12a^{2}z^{6}+a^{2}z^{4}+5z^{4}a^{-2}+6a^{2}z^{2}-2z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-3a^{2}-4a^{-2}+az^{9}+5az^{7}+2z^{7}a^{-1}-16az^{5}-z^{5}a^{-1}+13az^{3}+z^{3}a^{-1}+4za^{-1}-2az^{-1}-2a^{-1}z^{-1}+3z^{8}-6z^{6}+8z^{4}+z^{2}+2z^{-2}-6$ (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 χ 7                   1 -1 5                 4   4 3               4 3   -1 1             6 2     4 -1           5 5       0 -3         5 5         0 -5       3 5           2 -7     3 5             -2 -9   1 4               3 -11   2                 -2 -13 1                   1

Integral Khovanov Homology (db, data source)

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

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.

L11n308 L11n310