K11n48

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K11n47.gif

K11n47

K11n49.gif

K11n49

Contents

K11n48.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n48 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8394 X5,12,6,13 X7,17,8,16 X2,9,3,10 X11,19,12,18 X13,20,14,21 X15,22,16,1 X17,11,18,10 X19,7,20,6 X21,14,22,15
Gauss code 1, -5, 2, -1, -3, 10, -4, -2, 5, 9, -6, 3, -7, 11, -8, 4, -9, 6, -10, 7, -11, 8
Dowker-Thistlethwaite code 4 8 -12 -16 2 -18 -20 -22 -10 -6 -14
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation K11n48 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n48/ThurstonBennequinNumber
Hyperbolic Volume 9.99604
A-Polynomial See Data:K11n48/A-polynomial

[edit Notes for K11n48's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11n48's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+3 t^2-6 t+9-6 t^{-1} +3 t^{-2} - t^{-3}
Conway polynomial -z^6-3 z^4-3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 29, 0 }
Jones polynomial 2 q^2-3 q+4-5 q^{-1} +5 q^{-2} -4 q^{-3} +3 q^{-4} -2 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) -a^2 z^6+a^4 z^4-5 a^2 z^4+z^4+3 a^4 z^2-8 a^2 z^2+2 z^2+2 a^4-3 a^2+ a^{-2} +1
Kauffman polynomial (db, data sources) a^3 z^9+a z^9+2 a^4 z^8+3 a^2 z^8+z^8+2 a^5 z^7-3 a^3 z^7-5 a z^7+a^6 z^6-8 a^4 z^6-14 a^2 z^6-5 z^6-8 a^5 z^5+2 a^3 z^5+11 a z^5+z^5 a^{-1} -4 a^6 z^4+9 a^4 z^4+24 a^2 z^4+11 z^4+7 a^5 z^3-2 a^3 z^3-9 a z^3+3 a^6 z^2-6 a^4 z^2-17 a^2 z^2+2 z^2 a^{-2} -6 z^2-a^5 z+a^3 z+2 a z+2 a^4+3 a^2- a^{-2} +1
The A2 invariant q^{18}+q^{14}-q^{10}+q^8-q^6-q^2-1+ q^{-2} +2 q^{-6} + q^{-8}
The G2 invariant Data:K11n48/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: (-3, 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
-12 16 72 114 62 -192 -\frac{1088}{3} -\frac{320}{3} -80 -288 128 -1368 -744 -\frac{8831}{10} \frac{8026}{15} -\frac{21902}{15} \frac{1567}{6} -\frac{3871}{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=0 is the signature of K11n48. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012χ
5        22
3       1 -1
1      32 1
-1     32  -1
-3    22   0
-5   23    1
-7  12     -1
-9 12      1
-11 1       -1
-131        1
Integral Khovanov Homology

(db, data source)

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

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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K11n47.gif

K11n47

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K11n49