# L11a176

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## Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a176's page at Knotilus. Visit L11a176's page at the original Knot Atlas.

### Link Presentations

 Planar diagram presentation X8192 X18,9,19,10 X20,13,21,14 X10,4,11,3 X14,6,15,5 X16,8,17,7 X22,16,7,15 X4,12,5,11 X12,19,13,20 X6,21,1,22 X2,18,3,17 Gauss code {1, -11, 4, -8, 5, -10}, {6, -1, 2, -4, 8, -9, 3, -5, 7, -6, 11, -2, 9, -3, 10, -7}

### Polynomial invariants

 Multivariable Alexander Polynomial (in u, v, w, ...) $\frac{t(1)^2 t(2)^6-t(1) t(2)^6-3 t(1)^2 t(2)^5+4 t(1) t(2)^5-t(2)^5+5 t(1)^2 t(2)^4-8 t(1) t(2)^4+3 t(2)^4-5 t(1)^2 t(2)^3+11 t(1) t(2)^3-5 t(2)^3+3 t(1)^2 t(2)^2-8 t(1) t(2)^2+5 t(2)^2-t(1)^2 t(2)+4 t(1) t(2)-3 t(2)-t(1)+1}{t(1) t(2)^3}$ (db) Jones polynomial $-q^{13/2}+4 q^{11/2}-9 q^{9/2}+15 q^{7/2}-21 q^{5/2}+23 q^{3/2}-24 \sqrt{q}+\frac{20}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial −z7a−3−4z5a−3−5z3a−3−2za−3−a−3z−1 + z9a−1−az7 + 6z7a−1−4az5 + 13z5a−1−5az3 + 12z3a−1−2az + 5za−1 + a−1z−1 (db) Kauffman polynomial z5a−7−z3a−7 + 4z6a−6−5z4a−6 + z2a−6 + 8z7a−5−12z5a−5 + 6z3a−5−za−5 + 10z8a−4 + a4z6−15z6a−4−2a4z4 + 8z4a−4 + a4z2−z2a−4 + 8z9a−3 + 4a3z7−8z7a−3−9a3z5 + z5a−3 + 5a3z3 + 2za−3−a−3z−1 + 3z10a−2 + 7a2z8 + 10z8a−2−15a2z6−27z6a−2 + 8a2z4 + 19z4a−2−a2z2−4z2a−2 + a−2 + 7az9 + 15z9a−1−12az7−32z7a−1 + 6az5 + 29z5a−1−6az3−18z3a−1 + 3az + 6za−1−a−1z−1 + 3z10 + 7z8−24z6 + 16z4−4z2 (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 L11a176. 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      1210    -2
2     1211     1
0    913      4
-2   611       -5
-4  39        6
-6 16         -5
-8 3          3
-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 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −3 ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ r = −2 ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ r = −1 ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ r = 0 ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{12}$ r = 1 ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ r = 2 ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{12}$ 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.

###  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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