# K11a20

## Contents

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a20's page at Knotilus! Visit K11a20's page at the original Knot Atlas!

### Knot presentations

 Planar diagram presentation X4251 X8493 X12,6,13,5 X2837 X18,9,19,10 X6,12,7,11 X20,14,21,13 X22,16,1,15 X10,17,11,18 X16,20,17,19 X14,22,15,21 Gauss code 1, -4, 2, -1, 3, -6, 4, -2, 5, -9, 6, -3, 7, -11, 8, -10, 9, -5, 10, -7, 11, -8 Dowker-Thistlethwaite code 4 8 12 2 18 6 20 22 10 16 14
A Braid Representative

### Three dimensional invariants

 Symmetry type Chiral Unknotting number {2,3} 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a20/ThurstonBennequinNumber Hyperbolic Volume 15.3455 A-Polynomial See Data:K11a20/A-polynomial

### Four dimensional invariants

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

### Polynomial invariants

 Alexander polynomial −3t3 + 13t2−25t + 31−25t−1 + 13t−2−3t−3 Conway polynomial −3z6−5z4 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 113, 4 } Jones polynomial −q11 + 4q10−8q9 + 13q8−17q7 + 18q6−18q5 + 15q4−10q3 + 6q2−2q + 1 HOMFLY-PT polynomial (db, data sources) −z6a−4−2z6a−6 + z4a−2−2z4a−4−7z4a−6 + 3z4a−8 + 3z2a−2−9z2a−6 + 7z2a−8−z2a−10 + 2a−2 + a−4−5a−6 + 4a−8−a−10 Kauffman polynomial (db, data sources) z10a−6 + z10a−8 + 3z9a−5 + 7z9a−7 + 4z9a−9 + 3z8a−4 + 8z8a−6 + 12z8a−8 + 7z8a−10 + 2z7a−3−4z7a−5−8z7a−7 + 5z7a−9 + 7z7a−11 + z6a−2−6z6a−4−26z6a−6−30z6a−8−7z6a−10 + 4z6a−12−5z5a−3 + z5a−5−5z5a−7−23z5a−9−11z5a−11 + z5a−13−4z4a−2 + 3z4a−4 + 34z4a−6 + 30z4a−8−3z4a−10−6z4a−12 + 2z3a−3 + 14z3a−7 + 22z3a−9 + 5z3a−11−z3a−13 + 5z2a−2−2z2a−4−22z2a−6−15z2a−8 + 2z2a−10 + 2z2a−12 + za−3−za−5−7za−7−7za−9−2za−11−2a−2 + a−4 + 5a−6 + 4a−8 + a−10 The A2 invariant Data:K11a20/QuantumInvariant/A2/1,0 The G2 invariant Data:K11a20/QuantumInvariant/G2/1,0

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

Same Alexander/Conway Polynomial: {}

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

### Vassiliev invariants

 V2 and V3: (0, -1)
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 0 −8 0 32 40 0 $\frac{976}{3}$ $\frac{352}{3}$ 152 0 32 0 0 1184 8 $\frac{2264}{3}$ $\frac{416}{3}$ 16

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 = 4 is the signature of K11a20. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-2-10123456789χ
23           1-1
21          3 3
19         51 -4
17        83  5
15       95   -4
13      98    1
11     99     0
9    69      -3
7   49       5
5  26        -4
3 15         4
1 1          -1
-11           1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = 3 i = 5 r = −2 ${\mathbb Z}$ r = −1 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 0 ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ r = 1 ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ r = 2 ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ r = 3 ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ r = 4 ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ r = 5 ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ r = 6 ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ r = 7 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ r = 8 ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ r = 9 ${\mathbb Z}_2$ ${\mathbb Z}$

### 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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