# L7a1

## Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7a1's page at Knotilus. Visit L7a1's page at the original Knot Atlas. L7a1 is $7^2_6$ in the Rolfsen table of links.

 Planar diagram presentation X6172 X12,7,13,8 X4,13,1,14 X10,6,11,5 X8493 X14,10,5,9 X2,12,3,11 Gauss code {1, -7, 5, -3}, {4, -1, 2, -5, 6, -4, 7, -2, 3, -6}

### Polynomial invariants

 Multivariable Alexander Polynomial (in u, v, w, ...) $\frac{(u-1) (v-1) \left(v^2-v+1\right)}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $-q^{9/2}+3 q^{7/2}-4 q^{5/2}+\frac{1}{q^{5/2}}+4 q^{3/2}-\frac{3}{q^{3/2}}-5 \sqrt{q}+\frac{3}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial −z3a−3−za−3 + z5a−1−az3 + 3z3a−1−az + 2za−1 + az−1−a−1z−1 (db) Kauffman polynomial z3a−5 + 3z4a−4−2z2a−4 + 4z5a−3−5z3a−3 + 2za−3 + 2z6a−2 + a2z4 + z4a−2−a2z2−3z2a−2 + 3az5 + 7z5a−1−6az3−12z3a−1 + 2az + 4za−1 + az−1 + a−1z−1 + 2z6−z4−2z2−1 (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 L7a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
\ r
\
j \
-3-2-101234χ
10       11
8      2 -2
6     21 1
4    22  0
2   32   1
0  24    2
-2 11     0
-4 2      2
-61       -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = 0 i = 2 r = −3 ${\mathbb Z}$ r = −2 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −1 ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = 0 ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ r = 1 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = 2 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ 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).