L10n78

L10n77

L10n79

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n78 at Knotilus!

Link Presentations

[edit Notes on L10n78's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X9,18,10,19 X11,20,12,13 X13,12,14,5 X17,1,18,4 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 $u$, $v$, $w$, ...)	${\frac {t(3)^{2}t(2)^{3}-t(1)t(3)^{3}t(2)^{2}+t(1)t(3)^{2}t(2)^{2}-t(3)^{2}t(2)^{2}+t(3)t(2)^{2}-t(1)t(3)^{2}t(2)+t(1)t(3)t(2)-t(3)t(2)+t(2)-t(1)t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$2q^{-2}-q^{-3}+3q^{-4}-3q^{-5}+4q^{-6}-3q^{-7}+2q^{-8}-q^{-9}+q^{-10}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{8}z^{4}+4a^{8}z^{2}+a^{8}z^{-2}+4a^{8}-a^{6}z^{6}-6a^{6}z^{4}-13a^{6}z^{2}-2a^{6}z^{-2}-11a^{6}+2a^{4}z^{4}+8a^{4}z^{2}+a^{4}z^{-2}+7a^{4}$ (db)
Kauffman polynomial	$z^{4}a^{12}-3z^{2}a^{12}+a^{12}+z^{5}a^{11}-2z^{3}a^{11}+z^{6}a^{10}-2z^{4}a^{10}+z^{2}a^{10}+z^{7}a^{9}-3z^{5}a^{9}+4z^{3}a^{9}+z^{8}a^{8}-5z^{6}a^{8}+11z^{4}a^{8}-8z^{2}a^{8}-a^{8}z^{-2}+5a^{8}+2z^{7}a^{7}-9z^{5}a^{7}+17z^{3}a^{7}-11za^{7}+2a^{7}z^{-1}+z^{8}a^{6}-6z^{6}a^{6}+17z^{4}a^{6}-23z^{2}a^{6}-2a^{6}z^{-2}+13a^{6}+z^{7}a^{5}-5z^{5}a^{5}+11z^{3}a^{5}-11za^{5}+2a^{5}z^{-1}+3z^{4}a^{4}-11z^{2}a^{4}-a^{4}z^{-2}+8a^{4}$ (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               1 2 1 -7             2     2 -9           1 1     0 -11         3 2       1 -13       1 2         1 -15     1 2           -1 -17     1             1 -19 1 1               0 -21 1                 1

Integral Khovanov Homology (db, data source)

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

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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