L10n79

L10n78

L10n80

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n79 at Knotilus!

Link Presentations

[edit Notes on L10n79's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)(w-1)(vw+1)}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$2q^{-2}-2q^{-3}+5q^{-4}-5q^{-5}+6q^{-6}-5q^{-7}+4q^{-8}-2q^{-9}+q^{-10}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$z^{4}a^{8}+3z^{2}a^{8}+a^{8}z^{-2}+3a^{8}-z^{6}a^{6}-5z^{4}a^{6}-10z^{2}a^{6}-2a^{6}z^{-2}-9a^{6}+2z^{4}a^{4}+7z^{2}a^{4}+a^{4}z^{-2}+6a^{4}$ (db)
Kauffman polynomial	$a^{12}z^{4}-2a^{12}z^{2}+a^{12}+2a^{11}z^{5}-3a^{11}z^{3}+2a^{10}z^{6}-a^{10}z^{4}-2a^{10}z^{2}+2a^{9}z^{7}-3a^{9}z^{5}+3a^{9}z^{3}+a^{8}z^{8}-a^{8}z^{6}+3a^{8}z^{4}-3a^{8}z^{2}-a^{8}z^{-2}+3a^{8}+3a^{7}z^{7}-8a^{7}z^{5}+13a^{7}z^{3}-9a^{7}z+2a^{7}z^{-1}+a^{6}z^{8}-3a^{6}z^{6}+8a^{6}z^{4}-12a^{6}z^{2}-2a^{6}z^{-2}+9a^{6}+a^{5}z^{7}-3a^{5}z^{5}+7a^{5}z^{3}-9a^{5}z+2a^{5}z^{-1}+3a^{4}z^{4}-9a^{4}z^{2}-a^{4}z^{-2}+6a^{4}$ (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 -3                 2 2 -5               2 2 0 -7             3     3 -9           2 2     0 -11         4 3       1 -13       1 2         1 -15     3 4           -1 -17   1 3             2 -19   1               -1 -21 1                 1

Integral Khovanov Homology (db, data source)

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

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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