K11a228

Contents

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a228's page at Knotilus! Visit K11a228's page at the original Knot Atlas!

Knot presentations

 Planar diagram presentation X4251 X12,3,13,4 X18,5,19,6 X22,8,1,7 X20,9,21,10 X14,12,15,11 X2,13,3,14 X8,16,9,15 X6,17,7,18 X10,19,11,20 X16,22,17,21 Gauss code 1, -7, 2, -1, 3, -9, 4, -8, 5, -10, 6, -2, 7, -6, 8, -11, 9, -3, 10, -5, 11, -4 Dowker-Thistlethwaite code 4 12 18 22 20 14 2 8 6 10 16
A Braid Representative

Three dimensional invariants

 Symmetry type Reversible Unknotting number {1,2} 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a228/ThurstonBennequinNumber Hyperbolic Volume 16.809 A-Polynomial See Data:K11a228/A-polynomial

Four dimensional invariants

 Smooth 4 genus Missing Topological 4 genus Missing Concordance genus 3 Rasmussen s-Invariant 0

Polynomial invariants

 Alexander polynomial −t3 + 10t2−32t + 47−32t−1 + 10t−2−t−3 Conway polynomial −z6 + 4z4−z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 133, 0 } Jones polynomial −q5 + 4q4−9q3 + 15q2−19q + 22−21q−1 + 18q−2−13q−3 + 7q−4−3q−5 + q−6 HOMFLY-PT polynomial (db, data sources) a6−3z2a4−2a4 + 3z4a2 + 3z2a2 + a2−z6−z4−z2 + 1 + 2z4a−2 + z2a−2−z2a−4 Kauffman polynomial (db, data sources) 2a2z10 + 2z10 + 4a3z9 + 11az9 + 7z9a−1 + 4a4z8 + 6a2z8 + 10z8a−2 + 12z8 + 3a5z7−2a3z7−19az7−6z7a−1 + 8z7a−3 + a6z6−5a4z6−14a2z6−17z6a−2 + 4z6a−4−29z6−8a5z5−8a3z5 + 13az5−12z5a−3 + z5a−5−3a6z4−5a4z4 + 3a2z4 + 12z4a−2−5z4a−4 + 22z4 + 7a5z3 + 7a3z3−7az3−z3a−1 + 5z3a−3−z3a−5 + 3a6z2 + 8a4z2 + 3a2z2−4z2a−2 + z2a−4−7z2−2a5z−2a3z + az + za−1−a6−2a4−a2 + 1 The A2 invariant Data:K11a228/QuantumInvariant/A2/1,0 The G2 invariant Data:K11a228/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a251, K11a253,}

Vassiliev invariants

 V2 and V3: (-1, 2)
V2,1 through V6,9:
 V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 −4 16 8 $-\frac{158}{3}$ $-\frac{82}{3}$ −64 $\frac{64}{3}$ $-\frac{224}{3}$ 112 $-\frac{32}{3}$ 128 $\frac{632}{3}$ $\frac{328}{3}$ $\frac{11249}{30}$ $\frac{3022}{15}$ $\frac{4498}{45}$ $-\frac{2033}{18}$ $-\frac{271}{30}$

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

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 = 0 is the signature of K11a228. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-6-5-4-3-2-1012345χ
11           1-1
9          3 3
7         61 -5
5        93  6
3       106   -4
1      129    3
-1     1011     1
-3    811      -3
-5   510       5
-7  28        -6
-9 15         4
-11 2          -2
-131           1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −1 i = 1 r = −6 ${\mathbb Z}$ r = −5 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −4 ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = −3 ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ r = −2 ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ r = −1 ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ r = 0 ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{12}$ r = 1 ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ r = 2 ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ r = 3 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ r = 4 ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ r = 5 ${\mathbb Z}_2$ ${\mathbb Z}$

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 Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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