L10n75

L10n74

L10n76

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n75 at Knotilus!

Link Presentations

[edit Notes on L10n75's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X9,18,10,19 X17,1,18,4 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
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-uv^{2}w^{2}+uv^{2}w-uvw^{3}+2uvw^{2}-uvw-uw^{2}+v^{3}w+v^{2}w^{2}-2v^{2}w+v^{2}-vw^{2}+vw}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$2q^{-1}-3q^{-2}+5q^{-3}-4q^{-4}+5q^{-5}-4q^{-6}+3q^{-7}-q^{-8}+q^{-9}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z^{2}a^{8}+a^{8}z^{-2}+2a^{8}-z^{4}a^{6}-3z^{2}a^{6}-2a^{6}z^{-2}-5a^{6}-z^{4}a^{4}-z^{2}a^{4}+a^{4}z^{-2}+a^{4}+2z^{2}a^{2}+2a^{2}$ (db)
Kauffman polynomial	$z^{6}a^{10}-5z^{4}a^{10}+7z^{2}a^{10}-2a^{10}+z^{7}a^{9}-3z^{5}a^{9}+z^{3}a^{9}+za^{9}+z^{8}a^{8}-3z^{6}a^{8}+3z^{4}a^{8}-4z^{2}a^{8}-a^{8}z^{-2}+3a^{8}+3z^{7}a^{7}-9z^{5}a^{7}+9z^{3}a^{7}-8za^{7}+2a^{7}z^{-1}+z^{8}a^{6}-2z^{6}a^{6}+6z^{4}a^{6}-14z^{2}a^{6}-2a^{6}z^{-2}+9a^{6}+2z^{7}a^{5}-5z^{5}a^{5}+9z^{3}a^{5}-8za^{5}+2a^{5}z^{-1}+2z^{6}a^{4}-2z^{4}a^{4}-a^{4}z^{-2}+3a^{4}+z^{5}a^{3}+z^{3}a^{3}+za^{3}+3z^{2}a^{2}-2a^{2}$ (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           2 3     1 -9         3 2       1 -11       2 3         1 -13     1 2           -1 -15     2             2 -17 1 1               0 -19 1                 1

Integral Khovanov Homology (db, data source)

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

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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