# K11a52

Jump to: navigation, search

## Contents

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a52's page at Knotilus! Visit K11a52's page at the original Knot Atlas!

### Knot presentations

 Planar diagram presentation X4251 X8394 X14,5,15,6 X12,8,13,7 X2,9,3,10 X18,12,19,11 X22,13,1,14 X20,16,21,15 X10,18,11,17 X16,20,17,19 X6,21,7,22 Gauss code 1, -5, 2, -1, 3, -11, 4, -2, 5, -9, 6, -4, 7, -3, 8, -10, 9, -6, 10, -8, 11, -7 Dowker-Thistlethwaite code 4 8 14 12 2 18 22 20 10 16 6
A Braid Representative
A Morse Link Presentation

### Three dimensional invariants

 Symmetry type Chiral Unknotting number {1,2} 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a52/ThurstonBennequinNumber Hyperbolic Volume 16.8243 A-Polynomial See Data:K11a52/A-polynomial

### Four dimensional invariants

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

### Polynomial invariants

 Alexander polynomial −2t3 + 13t2−32t + 43−32t−1 + 13t−2−2t−3 Conway polynomial −2z6 + z4 + 2z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 137, 0 } Jones polynomial q6−5q5 + 10q4−15q3 + 20q2−22q + 22−18q−1 + 13q−2−7q−3 + 3q−4−q−5 HOMFLY-PT polynomial (db, data sources) −z6a−2−z6 + 2a2z4−z4a−2 + z4a−4−z4−a4z2 + 3a2z2 + z2a−2−z2−a4 + 2a2 + 2a−2−a−4−1 Kauffman polynomial (db, data sources) 2z10a−2 + 2z10 + 5az9 + 12z9a−1 + 7z9a−3 + 6a2z8 + 14z8a−2 + 9z8a−4 + 11z8 + 5a3z7−18z7a−1−8z7a−3 + 5z7a−5 + 3a4z6−5a2z6−41z6a−2−21z6a−4 + z6a−6−27z6 + a5z5−6a3z5−9az5−z5a−1−9z5a−3−10z5a−5−5a4z4−a2z4 + 27z4a−2 + 11z4a−4−z4a−6 + 19z4−2a5z3 + 2a3z3 + 10az3 + 10z3a−1 + 7z3a−3 + 3z3a−5 + 3a4z2 + 4a2z2−2z2a−2−z2 + a5z−3az−3za−1 + za−5−a4−2a2−2a−2−a−4−1 The A2 invariant Data:K11a52/QuantumInvariant/A2/1,0 The G2 invariant Data:K11a52/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: (2, 0)
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 0 32 $\frac{172}{3}$ $\frac{20}{3}$ 0 0 32 −32 $\frac{256}{3}$ 0 $\frac{1376}{3}$ $\frac{160}{3}$ $\frac{8551}{15}$ $\frac{1156}{15}$ $\frac{5404}{45}$ $\frac{377}{9}$ $-\frac{89}{15}$

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 = 0 is the signature of K11a52. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-5-4-3-2-10123456χ
13           11
11          4 -4
9         61 5
7        94  -5
5       116   5
3      119    -2
1     1111     0
-1    812      4
-3   510       -5
-5  28        6
-7 15         -4
-9 2          2
-111           -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −1 i = 1 r = −5 ${\mathbb Z}$ r = −4 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −3 ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = −2 ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ r = −1 ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ r = 0 ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{11}$ r = 1 ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ r = 2 ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ r = 3 ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ r = 4 ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ r = 5 ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ 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 Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

Back to the top.