K11n57

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K11n56.gif

K11n56

K11n58.gif

K11n58

Contents

K11n57.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n57 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X5,15,6,14 X2837 X9,17,10,16 X11,19,12,18 X13,20,14,21 X15,7,16,6 X17,11,18,10 X19,1,20,22 X21,12,22,13
Gauss code 1, -4, 2, -1, -3, 8, 4, -2, -5, 9, -6, 11, -7, 3, -8, 5, -9, 6, -10, 7, -11, 10
Dowker-Thistlethwaite code 4 8 -14 2 -16 -18 -20 -6 -10 -22 -12
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation K11n57 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n57/ThurstonBennequinNumber
Hyperbolic Volume 5.86054
A-Polynomial See Data:K11n57/A-polynomial

[edit Notes for K11n57's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant -6

[edit Notes for K11n57's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+3 t^3-2 t^2-t+3- t^{-1} -2 t^{-2} +3 t^{-3} - t^{-4}
Conway polynomial -z^8-5 z^6-4 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 7, 6 }
Jones polynomial -q^6+q^5-q^4+2 q^3-q^2+q
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-6} +z^6 a^{-4} -7 z^6 a^{-6} +z^6 a^{-8} +6 z^4 a^{-4} -16 z^4 a^{-6} +6 z^4 a^{-8} +10 z^2 a^{-4} -16 z^2 a^{-6} +9 z^2 a^{-8} -z^2 a^{-10} +5 a^{-4} -7 a^{-6} +4 a^{-8} - a^{-10}
Kauffman polynomial (db, data sources) z^9 a^{-5} +z^9 a^{-7} +z^8 a^{-4} +3 z^8 a^{-6} +2 z^8 a^{-8} -6 z^7 a^{-5} -5 z^7 a^{-7} +z^7 a^{-9} -7 z^6 a^{-4} -20 z^6 a^{-6} -13 z^6 a^{-8} +9 z^5 a^{-5} +3 z^5 a^{-7} -6 z^5 a^{-9} +16 z^4 a^{-4} +40 z^4 a^{-6} +24 z^4 a^{-8} -z^3 a^{-5} +7 z^3 a^{-7} +8 z^3 a^{-9} -15 z^2 a^{-4} -29 z^2 a^{-6} -16 z^2 a^{-8} -2 z^2 a^{-10} -3 z a^{-5} -5 z a^{-7} -3 z a^{-9} -z a^{-11} +5 a^{-4} +7 a^{-6} +4 a^{-8} + a^{-10}
The A2 invariant  q^{-4} + q^{-6} + q^{-8} +2 q^{-10} + q^{-12} - q^{-16} - q^{-18} -2 q^{-20} + q^{-26} + q^{-32} - q^{-34}
The G2 invariant Data:K11n57/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {5_2,}

Vassiliev invariants

V2 and V3: (2, 3)
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
8 24 32 \frac{412}{3} \frac{140}{3} 192 720 192 184 \frac{256}{3} 288 \frac{3296}{3} \frac{1120}{3} \frac{49711}{15} -\frac{1124}{15} \frac{90484}{45} \frac{833}{9} \frac{3631}{15}

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=6 is the signature of K11n57. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456χ
17       110
15      11 0
13     111 -1
11    121  0
9   11    0
7  111    1
5 12      1
3         0
11        1
Integral Khovanov Homology

(db, data source)

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

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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K11n56.gif

K11n56

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K11n58