# L7a5

## Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7a5's page at Knotilus. Visit L7a5's page at the original Knot Atlas. L7a5 is $7^2_2$ in the Rolfsen table of links.

 Planar diagram presentation X8192 X10,3,11,4 X12,6,13,5 X14,11,7,12 X4,14,5,13 X2738 X6,9,1,10 Gauss code {1, -6, 2, -5, 3, -7}, {6, -1, 7, -2, 4, -3, 5, -4}

### Polynomial invariants

 Multivariable Alexander Polynomial (in u, v, w, ...) $\frac{(u+v-1) (u v-u-v)}{u v}$ (db) Jones polynomial $-q^{3/2}+2 \sqrt{q}-\frac{3}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial −za5 + z3a3 + za3 + a3z−1 + z3a−az−1−za−1 (db) Kauffman polynomial −z4a6 + 2z2a6−2z5a5 + 5z3a5−3za5−z6a4 + 2z2a4−4z5a3 + 8z3a3−6za3 + a3z−1−z6a2−z4a2 + 2z2a2−a2−2z5a + 2z3a−2za + az−1−2z4 + 2z2−z3a−1 + za−1 (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 = -1 is the signature of L7a5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. Data:L7a5/KhovanovTable
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −2 i = 0 r = −5 ${\mathbb Z}$ r = −4 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −3 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −2 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ r = −1 ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = 0 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ r = 1 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 2 ${\mathbb Z}_2$ ${\mathbb Z}$

### 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).