L11n334

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Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n334's page at Knotilus. Visit L11n334's page at the original Knot Atlas.

Link Presentations

 Planar diagram presentation X6172 X11,16,12,17 X8493 X2,18,3,17 X5,14,6,15 X18,7,19,8 X15,12,16,5 X13,20,14,21 X9,13,10,22 X21,11,22,10 X4,19,1,20 Gauss code {1, -4, 3, -11}, {-5, -1, 6, -3, -9, 10, -2, 7}, {-8, 5, -7, 2, 4, -6, 11, 8, -10, 9}
A Braid Representative
A Morse Link Presentation

Polynomial invariants

 Multivariable Alexander Polynomial (in u, v, w, ...) 0 (db) Jones polynomial q3−2q2 + 2q−1 + 2q−1 + q−2 + 2q−4−2q−5 + 2q−6−q−7 (db) Signature -1 (db) HOMFLY-PT polynomial −z2a6−a6 + z4a4 + 3z2a4 + a4z−2 + 2a4−2a2z−2−a2−z4−3z2 + z−2−1 + z2a−2 + a−2 (db) Kauffman polynomial a7z7−5a7z5 + 6a7z3−2a7z + 2a6z8−11a6z6 + 16a6z4−9a6z2 + 3a6 + a5z9−4a5z7−2a5z5 + 12a5z3−7a5z + 3a4z8−20a4z6 + 37a4z4−27a4z2−a4z−2 + 9a4 + a3z9−5a3z7 + 13a3z3−10a3z + 2a3z−1 + 2a2z8−14a2z6 + z6a−2 + 26a2z4−4z4a−2−20a2z2 + 3z2a−2−2a2z−2 + 7a2−a−2 + 2az7 + 2z7a−1−12az5−9z5a−1 + 15az3 + 8z3a−1−6az−za−1 + 2az−1 + z8−4z6 + z4 + z2−z−2 + 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 L11n334. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
\ r
\
j \
-7-6-5-4-3-2-101234χ
7           11
5          1 -1
3         11 0
1       221  1
-1      131   1
-3     223    3
-5    241     1
-7   112      2
-9  121       0
-11 11         0
-13 1          1
-151           -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −3 i = −1 i = 1 r = −7 ${\mathbb Z}$ r = −6 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −5 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −4 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −3 ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = −2 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ r = −1 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 0 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ r = 1 ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = 2 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 3 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ 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.

 Modifying This Page

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