# K11a56

## Contents

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a56's page at Knotilus! Visit K11a56's page at the original Knot Atlas!

### Knot presentations

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

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a56/ThurstonBennequinNumber Hyperbolic Volume 14.5406 A-Polynomial See Data:K11a56/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 + 11t2−25t + 33−25t−1 + 11t−2−2t−3 Conway polynomial −2z6−z4 + z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 109, 0 } Jones polynomial −q5 + 3q4−7q3 + 12q2−15q + 18−17q−1 + 15q−2−11q−3 + 6q−4−3q−5 + q−6 HOMFLY-PT polynomial (db, data sources) −a2z6−z6 + a4z4−3a2z4 + 2z4a−2−z4 + 2a4z2−5a2z2 + 3z2a−2−z2a−4 + 2z2 + a4−3a2 + a−2−a−4 + 3 Kauffman polynomial (db, data sources) a2z10 + z10 + 3a3z9 + 6az9 + 3z9a−1 + 4a4z8 + 7a2z8 + 5z8a−2 + 8z8 + 3a5z7−a3z7−4az7 + 5z7a−1 + 5z7a−3 + a6z6−9a4z6−17a2z6−3z6a−2 + 3z6a−4−13z6−9a5z5−10a3z5−8az5−15z5a−1−7z5a−3 + z5a−5−3a6z4 + 5a4z4 + 13a2z4−4z4a−2−5z4a−4 + 6z4 + 8a5z3 + 12a3z3 + 10az3 + 11z3a−1 + 3z3a−3−2z3a−5 + 2a6z2−a4z2−8a2z2 + 4z2a−2 + 3z2a−4−4z2−2a5z−5a3z−5az−3za−1 + za−5 + a4 + 3a2−a−2−a−4 + 3 The A2 invariant Data:K11a56/QuantumInvariant/A2/1,0 The G2 invariant Data:K11a56/QuantumInvariant/G2/1,0

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

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

### 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 $\frac{62}{3}$ $\frac{34}{3}$ 64 $\frac{256}{3}$ $-\frac{128}{3}$ 48 $\frac{32}{3}$ 128 $\frac{248}{3}$ $\frac{136}{3}$ $\frac{11311}{30}$ $\frac{338}{15}$ $\frac{7262}{45}$ $\frac{689}{18}$ $\frac{271}{30}$

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 K11a56. 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          2 2
7         51 -4
5        72  5
3       85   -3
1      107    3
-1     89     1
-3    79      -2
-5   48       4
-7  27        -5
-9 14         3
-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 ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = −3 ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ r = −2 ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ r = −1 ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ r = 0 ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ r = 1 ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ r = 2 ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ r = 3 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ 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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