# K11a90

## Contents

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a90's page at Knotilus! Visit K11a90's page at the original Knot Atlas!

### Knot presentations

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

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a90/ThurstonBennequinNumber Hyperbolic Volume 12.5188 A-Polynomial See Data:K11a90/A-polynomial

### Four dimensional invariants

 Smooth 4 genus Missing Topological 4 genus Missing Concordance genus 3 Rasmussen s-Invariant 2

### Polynomial invariants

 Alexander polynomial 2t3−10t2 + 20t−23 + 20t−1−10t−2 + 2t−3 Conway polynomial 2z6 + 2z4−2z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 87, -2 } Jones polynomial −q4 + 3q3−5q2 + 9q−11 + 13q−1−14q−2 + 12q−3−9q−4 + 6q−5−3q−6 + q−7 HOMFLY-PT polynomial (db, data sources) z2a6 + a6−2z4a4−4z2a4−a4 + z6a2 + 2z4a2−a2 + z6 + 3z4 + 3z2 + 2−z4a−2−2z2a−2 Kauffman polynomial (db, data sources) a2z10 + z10 + 3a3z9 + 6az9 + 3z9a−1 + 5a4z8 + 6a2z8 + 3z8a−2 + 4z8 + 6a5z7 + a3z7−16az7−10z7a−1 + z7a−3 + 5a6z6−5a4z6−21a2z6−13z6a−2−24z6 + 3a7z5−9a5z5−14a3z5 + 10az5 + 8z5a−1−4z5a−3 + a8z4−6a6z4−2a4z4 + 18a2z4 + 17z4a−2 + 30z4−3a7z3 + 8a5z3 + 14a3z3−3az3−2z3a−1 + 4z3a−3−a8z2 + 3a6z2 + 5a4z2−7a2z2−7z2a−2−15z2−3a5z−3a3z + az + za−1−a6−a4 + a2 + 2 The A2 invariant Data:K11a90/QuantumInvariant/A2/1,0 The G2 invariant Data:K11a90/QuantumInvariant/G2/1,0

### "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a118,}

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

### Vassiliev invariants

 V2 and V3: (-2, 3)
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 −8 24 32 $-\frac{28}{3}$ $\frac{4}{3}$ −192 −240 −128 24 $-\frac{256}{3}$ 288 $\frac{224}{3}$ $-\frac{32}{3}$ $\frac{11729}{15}$ $\frac{268}{5}$ $\frac{16796}{45}$ $-\frac{401}{9}$ $\frac{929}{15}$

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 = -2 is the signature of K11a90. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-6-5-4-3-2-1012345χ
9           1-1
7          2 2
5         31 -2
3        62  4
1       53   -2
-1      86    2
-3     76     -1
-5    57      -2
-7   47       3
-9  25        -3
-11 14         3
-13 2          -2
-151           1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −3 i = −1 r = −6 ${\mathbb Z}$ r = −5 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −4 ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = −3 ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ r = −2 ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ r = −1 ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ r = 0 ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ r = 1 ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ r = 2 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ r = 3 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ r = 4 ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ 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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