L10n76

L10n75

L10n77

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n76 at Knotilus!

Link Presentations

[edit Notes on L10n76's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X8493 X2,16,3,15 X16,7,17,8 X9,18,10,19 X4,17,1,18 X19,12,20,5 X11,20,12,13 X13,10,14,11
Gauss code	{1, -4, 3, -7}, {-2, -1, 5, -3, -6, 10, -9, 8}, {-10, 2, 4, -5, 7, 6, -8, 9}

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

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

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