# K11a100

## Contents

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a100's page at Knotilus! Visit K11a100's page at the original Knot Atlas!

### Knot presentations

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

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a100/ThurstonBennequinNumber Hyperbolic Volume 16.1755 A-Polynomial See Data:K11a100/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 + 15t2−32t + 41−32t−1 + 15t−2−3t−3 Conway polynomial −3z6−3z4 + z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 141, 4 } Jones polynomial −q11 + 5q10−10q9 + 15q8−21q7 + 23q6−22q5 + 19q4−13q3 + 8q2−3q + 1 HOMFLY-PT polynomial (db, data sources) −z6a−4−2z6a−6 + z4a−2−z4a−4−6z4a−6 + 3z4a−8 + 2z2a−2 + 2z2a−4−7z2a−6 + 5z2a−8−z2a−10 + a−2 + 2a−4−3a−6 + a−8 Kauffman polynomial (db, data sources) z10a−6 + z10a−8 + 4z9a−5 + 9z9a−7 + 5z9a−9 + 5z8a−4 + 16z8a−6 + 21z8a−8 + 10z8a−10 + 3z7a−3 + z7a−5 + 2z7a−7 + 14z7a−9 + 10z7a−11 + z6a−2−10z6a−4−41z6a−6−44z6a−8−9z6a−10 + 5z6a−12−7z5a−3−17z5a−5−38z5a−7−44z5a−9−15z5a−11 + z5a−13−3z4a−2 + 7z4a−4 + 36z4a−6 + 27z4a−8−4z4a−10−5z4a−12 + 5z3a−3 + 18z3a−5 + 37z3a−7 + 30z3a−9 + 6z3a−11 + 3z2a−2−4z2a−4−15z2a−6−6z2a−8 + 2z2a−10−za−3−7za−5−10za−7−4za−9−a−2 + 2a−4 + 3a−6 + a−8 The A2 invariant Data:K11a100/QuantumInvariant/A2/1,0 The G2 invariant Data:K11a100/QuantumInvariant/G2/1,0

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

Same Alexander/Conway Polynomial: {K11a290,}

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

### Vassiliev invariants

 V2 and V3: (1, 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 4 0 8 $\frac{14}{3}$ $\frac{82}{3}$ 0 64 64 96 $\frac{32}{3}$ 0 $\frac{56}{3}$ $\frac{328}{3}$ $\frac{9151}{30}$ $-\frac{2542}{15}$ $\frac{30782}{45}$ $\frac{1313}{18}$ $\frac{1951}{30}$

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 = 4 is the signature of K11a100. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-2-10123456789χ
23           1-1
21          4 4
19         61 -5
17        94  5
15       126   -6
13      119    2
11     1112     1
9    811      -3
7   511       6
5  38        -5
3 16         5
1 2          -2
-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 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 0 ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ r = 1 ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ r = 2 ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ r = 3 ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ r = 4 ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ r = 5 ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ r = 6 ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ r = 7 ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ r = 8 ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ 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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