# L11a403

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## Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a403's page at Knotilus. Visit L11a403's page at the original Knot Atlas.

### Link Presentations

 Planar diagram presentation X6172 X12,4,13,3 X16,9,17,10 X14,8,15,7 X18,12,19,11 X20,15,21,16 X22,18,11,17 X10,19,5,20 X8,22,9,21 X2536 X4,14,1,13 Gauss code {1, -10, 2, -11}, {10, -1, 4, -9, 3, -8}, {5, -2, 11, -4, 6, -3, 7, -5, 8, -6, 9, -7}

### Polynomial invariants

 Multivariable Alexander Polynomial (in u, v, w, ...) $-\frac{u v^2 w^4-2 u v^2 w^3+2 u v^2 w^2-u v^2 w-2 u v w^4+5 u v w^3-5 u v w^2+4 u v w-u v+u w^4-3 u w^3+4 u w^2-3 u w+u-v^2 w^4+3 v^2 w^3-4 v^2 w^2+3 v^2 w-v^2+v w^4-4 v w^3+5 v w^2-5 v w+2 v+w^3-2 w^2+2 w-1}{\sqrt{u} v w^2}$ (db) Jones polynomial q7−4q6 + 9q5−15q4 + 20q3−22q2 + 23q−18 + 15q−1−8q−2 + 4q−3−q−4 (db) Signature 2 (db) HOMFLY-PT polynomial z6a−4 + 3z4a−4 + 3z2a−4 + a−4−z8a−2−5z6a−2−a2z4−10z4a−2−2a2z2−9z2a−2 + a2z−2 + a−2z−2−2a−2 + 2z6 + 7z4 + 7z2−2z−2 + 1 (db) Kauffman polynomial 2z10a−2 + 2z10 + 5az9 + 14z9a−1 + 9z9a−3 + 4a2z8 + 23z8a−2 + 15z8a−4 + 12z8 + a3z7−11az7−28z7a−1−2z7a−3 + 14z7a−5−14a2z6−77z6a−2−25z6a−4 + 9z6a−6−57z6−3a3z5−2az5−6z5a−1−30z5a−3−19z5a−5 + 4z5a−7 + 17a2z4 + 73z4a−2 + 15z4a−4−7z4a−6 + z4a−8 + 67z4 + 3a3z3 + 13az3 + 26z3a−1 + 27z3a−3 + 10z3a−5−z3a−7−8a2z2−30z2a−2−7z2a−4 + 2z2a−6−29z2−a3z−3az−7za−1−7za−3−2za−5 + 4a−2 + 2a−4 + 3−2az−1−2a−1z−1 + a2z−2 + a−2z−2 + 2z−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 = 2 is the signature of L11a403. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. Data:L11a403/KhovanovTable
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = 1 i = 3 r = −5 ${\mathbb Z}$ r = −4 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −3 ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ r = −2 ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ r = −1 ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ r = 0 ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{11}$ r = 1 ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ r = 2 ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{11}$ r = 3 ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ r = 4 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ r = 5 ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ 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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