# K11a281

## Contents

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a281's page at Knotilus! Visit K11a281's page at the original Knot Atlas!

### Knot presentations

 Planar diagram presentation X6271 X10,3,11,4 X16,6,17,5 X12,8,13,7 X20,10,21,9 X2,11,3,12 X18,13,19,14 X4,16,5,15 X22,17,1,18 X8,20,9,19 X14,21,15,22 Gauss code 1, -6, 2, -8, 3, -1, 4, -10, 5, -2, 6, -4, 7, -11, 8, -3, 9, -7, 10, -5, 11, -9 Dowker-Thistlethwaite code 6 10 16 12 20 2 18 4 22 8 14

### Three dimensional invariants

 Symmetry type Reversible Unknotting number {1,2} 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a281/ThurstonBennequinNumber Hyperbolic Volume 18.3111 A-Polynomial See Data:K11a281/A-polynomial

### Four dimensional invariants

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

### Polynomial invariants

 Alexander polynomial −t4 + 6t3−18t2 + 33t−39 + 33t−1−18t−2 + 6t−3−t−4 Conway polynomial −z8−2z6−2z4−z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 155, 2 } Jones polynomial q7−4q6 + 10q5−17q4 + 22q3−25q2 + 25q−21 + 16q−1−9q−2 + 4q−3−q−4 HOMFLY-PT polynomial (db, data sources) −z8a−2−5z6a−2 + z6a−4 + 2z6−a2z4−11z4a−2 + 3z4a−4 + 7z4−2a2z2−12z2a−2 + 4z2a−4 + 9z2−a2−5a−2 + 2a−4 + 5 Kauffman polynomial (db, data sources) 3z10a−2 + 3z10 + 6az9 + 18z9a−1 + 12z9a−3 + 4a2z8 + 24z8a−2 + 19z8a−4 + 9z8 + a3z7−15az7−41z7a−1−8z7a−3 + 17z7a−5−13a2z6−81z6a−2−33z6a−4 + 10z6a−6−51z6−3a3z5 + 6az5 + 12z5a−1−25z5a−3−24z5a−5 + 4z5a−7 + 15a2z4 + 71z4a−2 + 19z4a−4−7z4a−6 + z4a−8 + 59z4 + 3a3z3 + 7az3 + 12z3a−1 + 22z3a−3 + 14z3a−5−7a2z2−28z2a−2−7z2a−4 + 3z2a−6−25z2−a3z−3az−5za−1−6za−3−3za−5 + a2 + 5a−2 + 2a−4 + 5 The A2 invariant −q12 + q10−2q6 + 5q4−2q2 + 3 + 2q−2−4q−4 + 4q−6−6q−8 + 3q−10−q−12−2q−14 + 4q−16−2q−18 + q−20 The G2 invariant Data:K11a281/QuantumInvariant/G2/1,0

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

Same Alexander/Conway Polynomial: {K11a19, K11a25,}

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 $\frac{34}{3}$ $\frac{62}{3}$ 64 $\frac{512}{3}$ $\frac{224}{3}$ 48 $-\frac{32}{3}$ 128 $-\frac{136}{3}$ $-\frac{248}{3}$ $\frac{10289}{30}$ $\frac{314}{5}$ $-\frac{542}{45}$ $\frac{2287}{18}$ $-\frac{751}{30}$

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 K11a281. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-5-4-3-2-10123456χ
15           11
13          3 -3
11         71 6
9        103  -7
7       127   5
5      1310    -3
3     1212     0
1    1014      4
-1   611       -5
-3  310        7
-5 16         -5
-7 3          3
-91           -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = 1 i = 3 r = −5 ${\mathbb Z}$ r = −4 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −3 ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ r = −2 ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ r = −1 ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ r = 0 ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{12}$ r = 1 ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ r = 2 ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ r = 3 ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ r = 4 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ r = 5 ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ 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.

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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