K11a366

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K11a365.gif

K11a365

K11a367.gif

K11a367

Contents

K11a366.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

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Knot presentations

Planar diagram presentation X10,2,11,1 X14,4,15,3 X16,6,17,5 X18,8,19,7 X20,10,21,9 X22,12,1,11 X4,14,5,13 X2,16,3,15 X8,18,9,17 X6,20,7,19 X12,22,13,21
Gauss code 1, -8, 2, -7, 3, -10, 4, -9, 5, -1, 6, -11, 7, -2, 8, -3, 9, -4, 10, -5, 11, -6
Dowker-Thistlethwaite code 10 14 16 18 20 22 4 2 8 6 12
A Braid Representative
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A Morse Link Presentation K11a366 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3,4\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a366/ThurstonBennequinNumber
Hyperbolic Volume 12.7681
A-Polynomial See Data:K11a366/A-polynomial

[edit Notes for K11a366's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a366's four dimensional invariants]

Polynomial invariants

Alexander polynomial 8 t^2-20 t+25-20 t^{-1} +8 t^{-2}
Conway polynomial 8 z^4+12 z^2+1
2nd Alexander ideal (db, data sources) \{3,t+1\}
Determinant and Signature { 81, 4 }
Jones polynomial -q^{13}+q^{12}-4 q^{11}+7 q^{10}-9 q^9+13 q^8-13 q^7+12 q^6-10 q^5+7 q^4-3 q^3+q^2
HOMFLY-PT polynomial (db, data sources) z^4 a^{-4} +3 z^4 a^{-6} +3 z^4 a^{-8} +z^4 a^{-10} +z^2 a^{-4} +5 z^2 a^{-6} +6 z^2 a^{-8} +z^2 a^{-10} -z^2 a^{-12} +3 a^{-8} -2 a^{-12}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +3 z^9 a^{-9} +4 z^9 a^{-11} +z^9 a^{-13} +6 z^8 a^{-8} +2 z^8 a^{-10} -3 z^8 a^{-12} +z^8 a^{-14} +7 z^7 a^{-7} -2 z^7 a^{-9} -11 z^7 a^{-11} -z^7 a^{-13} +z^7 a^{-15} +6 z^6 a^{-6} -11 z^6 a^{-8} -3 z^6 a^{-10} +11 z^6 a^{-12} -3 z^6 a^{-14} +3 z^5 a^{-5} -10 z^5 a^{-7} +2 z^5 a^{-9} +15 z^5 a^{-11} -6 z^5 a^{-13} -6 z^5 a^{-15} +z^4 a^{-4} -8 z^4 a^{-6} +15 z^4 a^{-8} -4 z^4 a^{-10} -28 z^4 a^{-12} -2 z^3 a^{-5} +6 z^3 a^{-7} -6 z^3 a^{-9} -16 z^3 a^{-11} +10 z^3 a^{-13} +12 z^3 a^{-15} -z^2 a^{-4} +5 z^2 a^{-6} -12 z^2 a^{-8} +z^2 a^{-10} +23 z^2 a^{-12} +4 z^2 a^{-14} +4 z a^{-11} -4 z a^{-13} -8 z a^{-15} +3 a^{-8} -2 a^{-12}
The A2 invariant Data:K11a366/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a366/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: (12, 40)
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
48 320 1152 2872 504 15360 \frac{84032}{3} \frac{15104}{3} 4288 18432 51200 137856 24192 \frac{1380382}{5} -\frac{8344}{15} \frac{1816648}{15} \frac{5666}{3} \frac{84542}{5}

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 t^rq^j are shown, along with their alternating sums \chi (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 K11a366. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
27           1-1
25            0
23         41 -3
21        3   3
19       64   -2
17      73    4
15     66     0
13    67      -1
11   46       2
9  36        -3
7  4         4
513          -2
31           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}^{3}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=8 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=9 {\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=10 {\mathbb Z}
r=11 {\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).

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