# L10n78

## Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n78's page at Knotilus. Visit L10n78's page at the original Knot Atlas.

 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}

### 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 a8z4 + 4a8z2 + a8z−2 + 4a8−a6z6−6a6z4−13a6z2−2a6z−2−11a6 + 2a4z4 + 8a4z2 + a4z−2 + 7a4 (db) Kauffman polynomial z4a12−3z2a12 + a12 + z5a11−2z3a11 + z6a10−2z4a10 + z2a10 + z7a9−3z5a9 + 4z3a9 + z8a8−5z6a8 + 11z4a8−8z2a8−a8z−2 + 5a8 + 2z7a7−9z5a7 + 17z3a7−11za7 + 2a7z−1 + z8a6−6z6a6 + 17z4a6−23z2a6−2a6z−2 + 13a6 + z7a5−5z5a5 + 11z3a5−11za5 + 2a5z−1 + 3z4a4−11z2a4−a4z−2 + 8a4 (db)

### Khovanov Homology

 The coefficients of the monomials trqj are shown, along with their alternating sums χ (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j−2r = s + 1 or j−2r = s−1, where s = -4 is the signature of L10n78. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
\ r
\
j \
-8-7-6-5-4-3-2-10χ
-3        22
-5       121
-7      2  2
-9     11  0
-11    32   1
-13   12    1
-15  12     -1
-17  1      1
-1911       0
-211        1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ 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.

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).