K11n59

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K11n58.gif

K11n58

K11n60.gif

K11n60

Contents

K11n59.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n59 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X5,15,6,14 X2837 X16,9,17,10 X18,12,19,11 X20,14,21,13 X15,7,16,6 X22,17,1,18 X12,20,13,19 X10,22,11,21
Gauss code 1, -4, 2, -1, -3, 8, 4, -2, 5, -11, 6, -10, 7, 3, -8, -5, 9, -6, 10, -7, 11, -9
Dowker-Thistlethwaite code 4 8 -14 2 16 18 20 -6 22 12 10
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation K11n59 ML.gif

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:K11n59/ThurstonBennequinNumber
Hyperbolic Volume 11.895
A-Polynomial See Data:K11n59/A-polynomial

[edit Notes for K11n59's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -4

[edit Notes for K11n59's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+6 t^2-12 t+15-12 t^{-1} +6 t^{-2} - t^{-3}
Conway polynomial -z^6+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 53, 4 }
Jones polynomial -q^{11}+3 q^{10}-5 q^9+7 q^8-9 q^7+9 q^6-8 q^5+6 q^4-3 q^3+2 q^2
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-6} +2 z^4 a^{-4} -4 z^4 a^{-6} +2 z^4 a^{-8} +6 z^2 a^{-4} -7 z^2 a^{-6} +5 z^2 a^{-8} -z^2 a^{-10} +4 a^{-4} -5 a^{-6} +3 a^{-8} - a^{-10}
Kauffman polynomial (db, data sources) z^9 a^{-7} +z^9 a^{-9} +2 z^8 a^{-6} +5 z^8 a^{-8} +3 z^8 a^{-10} +z^7 a^{-5} +3 z^7 a^{-9} +4 z^7 a^{-11} -7 z^6 a^{-6} -16 z^6 a^{-8} -6 z^6 a^{-10} +3 z^6 a^{-12} -z^5 a^{-5} -4 z^5 a^{-7} -14 z^5 a^{-9} -10 z^5 a^{-11} +z^5 a^{-13} +3 z^4 a^{-4} +16 z^4 a^{-6} +24 z^4 a^{-8} +4 z^4 a^{-10} -7 z^4 a^{-12} +z^3 a^{-5} +11 z^3 a^{-7} +19 z^3 a^{-9} +7 z^3 a^{-11} -2 z^3 a^{-13} -8 z^2 a^{-4} -16 z^2 a^{-6} -12 z^2 a^{-8} -2 z^2 a^{-10} +2 z^2 a^{-12} -3 z a^{-5} -6 z a^{-7} -6 z a^{-9} -3 z a^{-11} +4 a^{-4} +5 a^{-6} +3 a^{-8} + a^{-10}
The A2 invariant 2 q^{-6} +3 q^{-10} + q^{-12} - q^{-14} + q^{-16} -3 q^{-18} - q^{-22} +2 q^{-26} - q^{-28} + q^{-30} - q^{-34}
The G2 invariant Data:K11n59/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n8,}

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

Vassiliev invariants

V2 and V3: (3, 6)
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
12 48 72 238 34 576 1248 192 176 288 1152 2856 408 \frac{66591}{10} \frac{214}{15} \frac{39662}{15} \frac{545}{6} \frac{3231}{10}

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 K11n59. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
0123456789χ
23         1-1
21        2 2
19       31 -2
17      42  2
15     53   -2
13    44    0
11   45     1
9  24      -2
7 14       3
512        -1
32         2
Integral Khovanov Homology

(db, data source)

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

Modifying This Page

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K11n58.gif

K11n58

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K11n60