# L6a5

## Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L6a5's page at Knotilus. Visit L6a5's page at the original Knot Atlas. L6a5 is $6^3_1$ in the Rolfsen table of links. It is a closed three-link chain.
 Stained glass window of Trinity symbol, Brazil French coat of arms. Russian coat of arms. Russian passport page-number decoration.

 Planar diagram presentation X6172 X10,3,11,4 X12,7,9,8 X8,11,5,12 X2536 X4,9,1,10 Gauss code {1, -5, 2, -6}, {5, -1, 3, -4}, {6, -2, 4, -3}

### Polynomial invariants

 Multivariable Alexander Polynomial (in u, v, w, ...) $\frac{t(2) t(1)+t(3) t(1)-t(1)-t(2)+t(2) t(3)-t(3)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)}}$ (db) Jones polynomial q−1−2q−2 + 3q−3−q−4 + 3q−5−q−6 + q−7 (db) Signature -2 (db) HOMFLY-PT polynomial a8z−2−2a6z−2−3a6 + 2a4z2 + a4z−2 + 3a4 + a2z2 (db) Kauffman polynomial z4a8−3z2a8−a8z−2 + 3a8 + z5a7−z3a7−3za7 + 2a7z−1 + 4z4a6−9z2a6−2a6z−2 + 5a6 + z5a5 + z3a5−3za5 + 2a5z−1 + 3z4a4−5z2a4−a4z−2 + 3a4 + 2z3a3 + z2a2 (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 = -2 is the signature of L6a5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. Data:L6a5/KhovanovTable
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −3 i = −1 r = −6 ${\mathbb Z}$ r = −5 ${\mathbb Z}_2$ ${\mathbb Z}$ r = −4 ${\mathbb Z}^{3}$ ${\mathbb Z}^{3}$ r = −3 ${\mathbb Z}$ r = −2 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −1 ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = 0 ${\mathbb Z}$ ${\mathbb Z}$

### 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).