# L10n85

## Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n85's page at Knotilus. Visit L10n85's page at the original Knot Atlas.
 Link L10n85. A graph, L10n85. A part of a link and a part of a graph.

 Planar diagram presentation X6172 X10,3,11,4 X20,14,15,13 X7,16,8,17 X15,8,16,9 X18,12,19,11 X12,20,13,19 X14,18,5,17 X2536 X4,9,1,10 Gauss code {1, -9, 2, -10}, {-5, 4, 8, -6, 7, -3}, {9, -1, -4, 5, 10, -2, 6, -7, 3, -8}

### Polynomial invariants

 Multivariable Alexander Polynomial (in u, v, w, ...) $-\frac{(w-1) \left(u v^2 w-u v w+2 u v-u-v^2 w+2 v w-v+1\right)}{\sqrt{u} v w}$ (db) Jones polynomial q4 + 2q−4−3q3−3q−3 + 5q2 + 6q−2−6q−6q−1 + 8 (db) Signature 0 (db) HOMFLY-PT polynomial −z6 + a2z4 + z4a−2−4z4 + a2z2 + 2z2a−2−5z2 + a4−2a2 + a−2 + a4z−2−2a2z−2 + z−2 (db) Kauffman polynomial 3a4z4 + z4a−4−7a4z2−z2a−4−a4z−2 + 4a4 + a3z7−a3z5 + 3z5a−3 + 2a3z3−4z3a−3−5a3z + 2a3z−1 + a2z8−a2z6 + 4z6a−2 + 4a2z4−6z4a−2−8a2z2 + 3z2a−2−2a2z−2 + 6a2−a−2 + 4az7 + 3z7a−1−7az5−3z5a−1 + 7az3 + z3a−1−5az + 2az−1 + z8 + 3z6−6z4 + 3z2−z−2 + 2 (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 = 0 is the signature of L10n85. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
\ r
\
j \
-4-3-2-101234χ
9        11
7       2 -2
5      31 2
3     43  -1
1    42   2
-1   35    2
-3  33     0
-5 14      3
-712       -1
-92        2
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −1 i = 1 r = −4 ${\mathbb Z}^{2}$ ${\mathbb Z}$ r = −3 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −2 ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ r = −1 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ r = 0 ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ r = 1 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ r = 2 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ r = 3 ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ 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.

Read me first: Modifying Knot Pages

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