L10n66

L10n65

L10n67

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n66 at Knotilus!

Link Presentations

[edit Notes on L10n66's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X16,12,17,11 X18,13,19,14 X20,18,9,17 X12,19,13,20 X15,8,16,5 X7,14,8,15 X2536 X9,1,10,4
Gauss code	{1, -9, -2, 10}, {9, -1, -8, 7}, {-10, 2, 3, -6, 4, 8, -7, -3, 5, -4, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-uvw^{2}+uvw+uw^{3}-2uw^{2}+2vw-v-w^{2}+w}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$q^{-6}-q^{-5}+2q^{-4}+q^{3}-q^{-3}-q^{2}+q^{-2}+q+1$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-2a^{4}z^{2}-3a^{4}z^{-2}-5a^{4}+a^{2}z^{4}+5a^{2}z^{2}+4a^{2}z^{-2}+z^{2}a^{-2}+a^{-2}z^{-2}+8a^{2}+2a^{-2}-z^{4}-5z^{2}-3z^{-2}-6$ (db)
Kauffman polynomial	$a^{5}z^{7}+a^{3}z^{7}+az^{7}+z^{7}a^{-1}+a^{6}z^{6}+3a^{4}z^{6}+3a^{2}z^{6}+z^{6}a^{-2}+2z^{6}-4a^{5}z^{5}-5a^{3}z^{5}-6az^{5}-5z^{5}a^{-1}-5a^{6}z^{4}-16a^{4}z^{4}-20a^{2}z^{4}-5z^{4}a^{-2}-14z^{4}+2a^{5}z^{3}+4a^{3}z^{3}+6az^{3}+4z^{3}a^{-1}+7a^{6}z^{2}+23a^{4}z^{2}+35a^{2}z^{2}+6z^{2}a^{-2}+25z^{2}+a^{5}z+a^{3}z+az+za^{-1}-4a^{6}-14a^{4}-21a^{2}-4a^{-2}-14-a^{5}z^{-1}-a^{3}z^{-1}-az^{-1}-a^{-1}z^{-1}+a^{6}z^{-2}+3a^{4}z^{-2}+4a^{2}z^{-2}+a^{-2}z^{-2}+3z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                     1 1 5                       0 3                 1 1   0 1             3 1       2 -1           1 3 1       1 -3         1 2 2         1 -5       1 1             0 -7     1 1 1             1 -9   1 2                 1 -11                       0 -13 1                     1

Integral Khovanov Homology (db, data source)

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

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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