# L11a1

## Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a1's page at Knotilus. Visit L11a1's page at the original Knot Atlas.

 Link L11a1. A graph, link L11a1. A part of a link and a part of a graph.

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

### Polynomial invariants

 Multivariable Alexander Polynomial (in u, v, w, ...) $-\frac{(u-1) (v-1) \left(v^4-5 v^3+7 v^2-5 v+1\right)}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $-q^{13/2}+4 q^{11/2}-9 q^{9/2}+15 q^{7/2}-21 q^{5/2}+24 q^{3/2}-25 \sqrt{q}+\frac{21}{\sqrt{q}}-\frac{16}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial −z7a−1 + 2az5−3z5a−1 + 2z5a−3−a3z3 + 3az3−4z3a−1 + 4z3a−3−z3a−5−az + 2za−3−za−5 + a3z−1−2az−1 + 2a−1z−1−a−3z−1 (db) Kauffman polynomial −2z10a−2−2z10−7az9−13z9a−1−6z9a−3−9a2z8−17z8a−2−9z8a−4−17z8−5a3z7 + 5az7 + 12z7a−1−6z7a−3−8z7a−5−a4z6 + 20a2z6 + 36z6a−2 + 9z6a−4−4z6a−6 + 44z6 + 10a3z5 + 14az5 + 16z5a−1 + 25z5a−3 + 12z5a−5−z5a−7 + a4z4−11a2z4−19z4a−2−z4a−4 + 5z4a−6−25z4−4a3z3−9az3−14z3a−1−18z3a−3−8z3a−5 + z3a−7 + 2z2a−2−z2a−4−2z2a−6 + z2−a3z−4az−2za−1 + 3za−3 + 2za−5 + 1 + a3z−1 + 2az−1 + 2a−1z−1 + a−3z−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 L11a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
\ r
\
j \
-5-4-3-2-10123456χ
14           11
12          3 -3
10         61 5
8        93  -6
6       126   6
4      129    -3
2     1312     1
0    1014      4
-2   611       -5
-4  410        6
-6 16         -5
-8 4          4
-101           -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = 0 i = 2 r = −5 ${\mathbb Z}$ r = −4 ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −3 ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ r = −2 ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ r = −1 ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ r = 0 ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{13}$ r = 1 ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ r = 2 ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ 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.

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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