# L11n421

## Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n421's page at Knotilus. Visit L11n421's page at the original Knot Atlas.

 Planar diagram presentation X8192 X16,8,17,7 X3,10,4,11 X17,2,18,3 X9,19,10,18 X20,12,21,11 X14,6,15,5 X22,15,13,16 X6,14,1,13 X4,19,5,20 X12,22,7,21 Gauss code {1, 4, -3, -10, 7, -9}, {2, -1, -5, 3, 6, -11}, {9, -7, 8, -2, -4, 5, 10, -6, 11, -8}

### Polynomial invariants

 Multivariable Alexander Polynomial (in u, v, w, ...) $\frac{t(1) t(2)^2 t(3)^3-t(2)^2 t(3)^3-2 t(1) t(2)^2 t(3)^2+t(2)^2 t(3)^2+t(1) t(2) t(3)^2-t(1)^2 t(3)+2 t(1) t(3)-t(1) t(2) t(3)+t(1)^2-t(1)}{t(1) t(2) t(3)^{3/2}}$ (db) Jones polynomial q5−q4 + 3q3−3q2 + 4q−3 + 4q−1−2q−2 + 2q−3−q−4 (db) Signature 2 (db) HOMFLY-PT polynomial z2a−4 + a−4z−2 + 3a−4−a2z4−2z4a−2−3a2z2−8z2a−2−2a−2z−2−a2−8a−2 + z6 + 5z4 + 8z2 + z−2 + 6 (db) Kauffman polynomial z2a−6−a−6 + z3a−5 + 2z4a−4−4z2a−4−a−4z−2 + 4a−4 + a3z7 + z7a−3−5a3z5−4z5a−3 + 6a3z3 + 7z3a−3−a3z−6za−3 + 2a−3z−1 + 2a2z8 + 2z8a−2−11a2z6−11z6a−2 + 18a2z4 + 23z4a−2−11a2z2−25z2a−2−2a−2z−2 + 2a2 + 12a−2 + az9 + z9a−1−3az7−3z7a−1−3az5−2z5a−1 + 9az3 + 9z3a−1−3az−8za−1 + 2a−1z−1 + 4z8−22z6 + 39z4−31z2−z−2 + 10 (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 L11n421. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
\ r
\
j \
-5-4-3-2-101234χ
11         11
9        110
7       2  2
5      22  0
3     21   1
1    23    1
-1   21     1
-3  13      2
-5 11       0
-7 1        1
-91         -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = 1 i = 3 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 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ r = 1 ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = 2 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ r = 3 ${\mathbb Z}_2$ ${\mathbb Z}$ r = 4 ${\mathbb Z}$ ${\mathbb Z}$

### 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).