# L11a50

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## Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a50's page at Knotilus. Visit L11a50's page at the original Knot Atlas.

### Link Presentations

 Planar diagram presentation X6172 X18,7,19,8 X4,19,1,20 X12,6,13,5 X10,4,11,3 X22,14,5,13 X14,22,15,21 X20,12,21,11 X16,9,17,10 X2,16,3,15 X8,17,9,18 Gauss code {1, -10, 5, -3}, {4, -1, 2, -11, 9, -5, 8, -4, 6, -7, 10, -9, 11, -2, 3, -8, 7, -6}
A Braid Representative
A Morse Link Presentation

### Polynomial invariants

 Multivariable Alexander Polynomial (in u, v, w, ...) $\frac{2 (u-1) (v-1) \left(2 v^2-3 v+2\right)}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $-7 q^{9/2}+\frac{1}{q^{9/2}}+11 q^{7/2}-\frac{4}{q^{7/2}}-15 q^{5/2}+\frac{7}{q^{5/2}}+18 q^{3/2}-\frac{12}{q^{3/2}}-q^{13/2}+3 q^{11/2}-18 \sqrt{q}+\frac{15}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial −z3a−5−za−5−a−5z−1 + z5a−3−a3z3 + z3a−3 + 2za−3 + a3z−1 + 2a−3z−1 + az5 + 2z5a−1 + 3z3a−1−2az + za−1−az−1−a−1z−1 (db) Kauffman polynomial z5a−7−2z3a−7 + 3z6a−6−5z4a−6 + 6z7a−5−14z5a−5 + 12z3a−5−5za−5 + a−5z−1 + 7z8a−4 + a4z6−17z6a−4−2a4z4 + 19z4a−4−8z2a−4 + a−4 + 5z9a−3 + 4a3z7−7z7a−3−11a3z5−z5a−3 + 6a3z3 + 15z3a−3 + a3z−11za−3−a3z−1 + 2a−3z−1 + 2z10a−2 + 6a2z8 + 5z8a−2−16a2z6−22z6a−2 + 10a2z4 + 31z4a−2−2a2z2−15z2a−2 + a2 + 3a−2 + 5az9 + 10z9a−1−10az7−27z7a−1 + 4az5 + 29z5a−1−3az3−8z3a−1 + 3az−4za−1−az−1 + a−1z−1 + 2z10 + 4z8−19z6 + 19z4−9z2 + 2 (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 L11a50. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. Data:L11a50/KhovanovTable
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = 0 i = 2 r = −5 ${\mathbb Z}$ r = −4 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −3 ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ r = −2 ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ r = −1 ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ r = 0 ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{9}$ r = 1 ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ r = 2 ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ r = 3 ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ r = 4 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ r = 5 ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = 6 ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.

###  Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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