L11a318

Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a318's page at Knotilus. Visit L11a318's page at the original Knot Atlas.

 Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X8,9,1,10 X20,13,21,14 X16,8,17,7 X18,6,19,5 X6,18,7,17 X4,20,5,19 X22,15,9,16 X14,21,15,22 Gauss code {1, -2, 3, -9, 7, -8, 6, -4}, {4, -1, 2, -3, 5, -11, 10, -6, 8, -7, 9, -5, 11, -10}

Polynomial invariants

 Multivariable Alexander Polynomial (in u, v, w, ...) $\frac{(t(1)-1) (t(2)-1) \left(t(1) t(2)^4+t(1)^2 t(2)^3+t(1) t(2)^3+t(2)^3+t(1) t(2)^2+t(1)^2 t(2)+t(1) t(2)+t(2)+t(1)\right)}{t(1)^{3/2} t(2)^{5/2}}$ (db) Jones polynomial $-\frac{6}{q^{9/2}}+\frac{5}{q^{7/2}}-\frac{5}{q^{5/2}}-q^{3/2}+\frac{4}{q^{3/2}}+\frac{1}{q^{19/2}}-\frac{2}{q^{17/2}}+\frac{3}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{5}{q^{11/2}}+\sqrt{q}-\frac{3}{\sqrt{q}}$ (db) Signature -5 (db) HOMFLY-PT polynomial −z5a7−4z3a7−3za7 + z7a5 + 5z5a5 + 6z3a5−a5z−1 + z7a3 + 6z5a3 + 12z3a3 + 10za3 + 3a3z−1−z5a−5z3a−7za−2az−1 (db) Kauffman polynomial a12z2 + 2a11z3 + 3a10z4−2a10z2 + 4a9z5−6a9z3 + 5a8z6−12a8z4 + 3a8z2 + 6a7z7−22a7z5 + 20a7z3−6a7z + 5a6z8−21a6z6 + 23a6z4−8a6z2 + a6 + 3a5z9−12a5z7 + 8a5z5 + 5a5z3−a5z−1 + a4z10−a4z8−16a4z6 + 37a4z4−21a4z2 + 3a4 + 4a3z9−26a3z7 + 57a3z5−51a3z3 + 19a3z−3a3z−1 + a2z10−6a2z8 + 10a2z6−a2z4−7a2z2 + 3a2 + az9−8az7 + 23az5−28az3 + 13az−2az−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 = -5 is the signature of L11a318. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
\ r
\
j \
-7-6-5-4-3-2-101234χ
4           11
2            0
0         31 2
-2        1   -1
-4       43   1
-6      33    0
-8     32     1
-10    23      1
-12   23       -1
-14  12        1
-16 12         -1
-18 1          1
-201           -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −6 i = −4 r = −7 ${\mathbb Z}$ r = −6 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −5 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −4 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = −3 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = −2 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ r = −1 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ r = 0 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ r = 1 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 2 ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ r = 3 ${\mathbb Z}$ 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.

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