L10n71

L10n70

L10n72

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n71 at Knotilus!

Link Presentations

[edit Notes on L10n71's Link Presentations]

Planar diagram presentation	X6172 X3,12,4,13 X7,17,8,16 X9,11,10,20 X11,18,12,19 X15,9,16,8 X19,5,20,10 X17,14,18,15 X2536 X13,4,14,1
Gauss code	{1, -9, -2, 10}, {9, -1, -3, 6, -4, 7}, {-5, 2, -10, 8, -6, 3, -8, 5, -7, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(3)^{2}t(2)^{2}+t(1)t(2)^{2}-2t(1)t(3)t(2)^{2}-t(2)^{2}-2t(1)t(3)^{2}t(2)+2t(1)t(3)t(2)-2t(3)t(2)+2t(2)+t(1)t(3)^{2}-t(3)^{2}+2t(3)-1}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{3}-2q^{2}+5q-4+7q^{-1}-6q^{-2}+5q^{-3}-4q^{-4}+2q^{-5}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}-a^{4}z^{4}-3a^{4}z^{2}-2a^{4}+a^{2}z^{6}+4a^{2}z^{4}+5a^{2}z^{2}+a^{2}z^{-2}+z^{2}a^{-2}+a^{-2}z^{-2}+3a^{2}+2a^{-2}-2z^{4}-6z^{2}-2z^{-2}-4$ (db)
Kauffman polynomial	$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+5az^{7}+2z^{7}a^{-1}+3a^{4}z^{6}+4a^{2}z^{6}+z^{6}a^{-2}+2z^{6}+a^{5}z^{5}-6a^{3}z^{5}-13az^{5}-6z^{5}a^{-1}-5a^{4}z^{4}-17a^{2}z^{4}-4z^{4}a^{-2}-16z^{4}+3a^{5}z^{3}+7a^{3}z^{3}+6az^{3}+2z^{3}a^{-1}+3a^{6}z^{2}+6a^{4}z^{2}+14a^{2}z^{2}+6z^{2}a^{-2}+17z^{2}-2a^{5}z-6a^{3}z+4za^{-1}-a^{6}-a^{4}-3a^{2}-4a^{-2}-6-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 χ 7                 1 1 5               1   -1 3             4 1   3 1           2 3     1 -1         5 2       3 -3       2 3         1 -5     3 4           -1 -7   1 2             1 -9 1 3               -2 -11 2                 2

Integral Khovanov Homology (db, data source)

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

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.

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