# L11a8

## Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a8's page at Knotilus. Visit L11a8's page at the original Knot Atlas.

 Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X20,9,21,10 X8493 X18,22,19,21 X14,12,15,11 X12,5,13,6 X22,13,5,14 X10,19,11,20 X2,16,3,15 Gauss code {1, -11, 5, -3}, {8, -1, 2, -5, 4, -10, 7, -8, 9, -7, 11, -2, 3, -6, 10, -4, 6, -9}

### Polynomial invariants

 Multivariable Alexander Polynomial (in u, v, w, ...) $\frac{(t(1)-1) (t(2)-1) \left(t(2)^4-5 t(2)^3+9 t(2)^2-5 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $-q^{7/2}+5 q^{5/2}-11 q^{3/2}+17 \sqrt{q}-\frac{25}{\sqrt{q}}+\frac{27}{q^{3/2}}-\frac{27}{q^{5/2}}+\frac{23}{q^{7/2}}-\frac{17}{q^{9/2}}+\frac{10}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial −za7 + 3z3a5 + 2za5−a5z−1−3z5a3−5z3a3 + 3a3z−1 + z7a + 3z5a + 4z3a−za−2az−1−z5a−1−z3a−1 (db) Kauffman polynomial a8z6−2a8z4 + a8z2 + 4a7z7−8a7z5 + 6a7z3−2a7z + 8a6z8−17a6z6 + 14a6z4−6a6z2 + a6 + 7a5z9−3a5z7−18a5z5 + 18a5z3−4a5z−a5z−1 + 2a4z10 + 22a4z8−59a4z6 + 47a4z4−16a4z2 + 3a4 + 15a3z9−8a3z7−35a3z5 + z5a−3 + 30a3z3−a3z−3a3z−1 + 2a2z10 + 27a2z8−61a2z6 + 5z6a−2 + 39a2z4−4z4a−2−11a2z2 + 3a2 + 8az9 + 10az7 + 11z7a−1−41az5−15z5a−1 + 25az3 + 7z3a−1 + az−2az−1 + 13z8−15z6 + 4z4−2z2 (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 L11a8. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
\ r
\
j \
-7-6-5-4-3-2-101234χ
8           11
6          4 -4
4         71 6
2        104  -6
0       157   8
-2      1412    -2
-4     1313     0
-6    1014      4
-8   713       -6
-10  310        7
-12 17         -6
-14 3          3
-161           -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −2 i = 0 r = −7 ${\mathbb Z}$ r = −6 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −5 ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ r = −4 ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ r = −3 ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ r = −2 ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ r = −1 ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14}$ ${\mathbb Z}^{14}$ r = 0 ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{15}$ r = 1 ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ r = 2 ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ r = 3 ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ 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).