# K11a50

## Contents

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a50's page at Knotilus! Visit K11a50's page at the original Knot Atlas!

### Knot presentations

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

### Three dimensional invariants

 Symmetry type Reversible Unknotting number {1,2} 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a50/ThurstonBennequinNumber Hyperbolic Volume 13.4737 A-Polynomial See Data:K11a50/A-polynomial

### Four dimensional invariants

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

### Polynomial invariants

 Alexander polynomial −5t2 + 21t−31 + 21t−1−5t−2 Conway polynomial −5z4 + z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 83, 2 } Jones polynomial −q10 + 3q9−6q8 + 9q7−11q6 + 13q5−13q4 + 11q3−8q2 + 5q−2 + q−1 HOMFLY-PT polynomial (db, data sources) −z4a−2−2z4a−4−2z4a−6−z2a−4−2z2a−6 + 3z2a−8 + z2−a−6 + 2a−8−a−10 + 1 Kauffman polynomial (db, data sources) z10a−6 + z10a−8 + 3z9a−5 + 6z9a−7 + 3z9a−9 + 4z8a−4 + 6z8a−6 + 5z8a−8 + 3z8a−10 + 4z7a−3−2z7a−5−15z7a−7−8z7a−9 + z7a−11 + 3z6a−2−5z6a−4−24z6a−6−28z6a−8−12z6a−10 + 2z5a−1−5z5a−3−7z5a−5 + 3z5a−7−z5a−9−4z5a−11−2z4a−2 + 2z4a−4 + 23z4a−6 + 32z4a−8 + 14z4a−10 + z4−2z3a−1 + 5z3a−3 + 10z3a−5 + 7z3a−7 + 9z3a−9 + 5z3a−11 + z2a−4−8z2a−6−13z2a−8−6z2a−10−2z2−2za−3−4za−5−3za−7−3za−9−2za−11 + a−6 + 2a−8 + a−10 + 1 The A2 invariant Data:K11a50/QuantumInvariant/A2/1,0 The G2 invariant Data:K11a50/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: (1, 4)
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 32 8 $\frac{542}{3}$ $\frac{130}{3}$ 128 $\frac{2432}{3}$ $\frac{416}{3}$ 192 $\frac{32}{3}$ 512 $\frac{2168}{3}$ $\frac{520}{3}$ $\frac{106831}{30}$ $-\frac{5182}{15}$ $\frac{90302}{45}$ $\frac{2225}{18}$ $\frac{7951}{30}$

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 = 2 is the signature of K11a50. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-2-10123456789χ
21           1-1
19          2 2
17         41 -3
15        52  3
13       64   -2
11      75    2
9     66     0
7    57      -2
5   36       3
3  25        -3
1 14         3
-1 1          -1
-31           1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = 1 i = 3 r = −2 ${\mathbb Z}$ r = −1 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 0 ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ r = 1 ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ r = 2 ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ r = 3 ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ r = 4 ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ r = 5 ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ r = 6 ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ r = 7 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ r = 8 ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ 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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