K11n133

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K11n132.gif

K11n132

K11n134.gif

K11n134

Contents

K11n133.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n133 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X5,19,6,18 X7,20,8,21 X2,10,3,9 X11,17,12,16 X13,6,14,7 X15,9,16,8 X17,1,18,22 X19,15,20,14 X21,12,22,13
Gauss code 1, -5, 2, -1, -3, 7, -4, 8, 5, -2, -6, 11, -7, 10, -8, 6, -9, 3, -10, 4, -11, 9
Dowker-Thistlethwaite code 4 10 -18 -20 2 -16 -6 -8 -22 -14 -12
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation K11n133 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n133/ThurstonBennequinNumber
Hyperbolic Volume 10.992
A-Polynomial See Data:K11n133/A-polynomial

[edit Notes for K11n133's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n133's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-4 t^3+6 t^2-2 t-1-2 t^{-1} +6 t^{-2} -4 t^{-3} + t^{-4}
Conway polynomial z^8+4 z^6+2 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{5,t+1\}
Determinant and Signature { 25, 4 }
Jones polynomial -q^7+2 q^6-3 q^5+4 q^4-4 q^3+4 q^2-3 q+3- q^{-1}
HOMFLY-PT polynomial (db, data sources) z^8 a^{-4} -z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -4 z^4 a^{-2} +11 z^4 a^{-4} -5 z^4 a^{-6} -2 z^2 a^{-2} +8 z^2 a^{-4} -5 z^2 a^{-6} +z^2 a^{-8} + a^{-2} + a^{-4} - a^{-6}
Kauffman polynomial (db, data sources) 2 z^9 a^{-3} +2 z^9 a^{-5} +3 z^8 a^{-2} +6 z^8 a^{-4} +3 z^8 a^{-6} +z^7 a^{-1} -8 z^7 a^{-3} -8 z^7 a^{-5} +z^7 a^{-7} -15 z^6 a^{-2} -31 z^6 a^{-4} -16 z^6 a^{-6} -4 z^5 a^{-1} +3 z^5 a^{-3} +2 z^5 a^{-5} -5 z^5 a^{-7} +19 z^4 a^{-2} +42 z^4 a^{-4} +23 z^4 a^{-6} +3 z^3 a^{-1} +7 z^3 a^{-3} +10 z^3 a^{-5} +6 z^3 a^{-7} -6 z^2 a^{-2} -16 z^2 a^{-4} -11 z^2 a^{-6} -z^2 a^{-8} -z a^{-1} -3 z a^{-3} -5 z a^{-5} -3 z a^{-7} - a^{-2} + a^{-4} + a^{-6}
The A2 invariant -q^2+1+ q^{-2} + q^{-4} + q^{-6} + q^{-8} + q^{-10} -2 q^{-12} + q^{-14} - q^{-16} + q^{-18} -2 q^{-26} + q^{-28}
The G2 invariant Data:K11n133/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n50, K11n132,}

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{220}{3} -\frac{4}{3} 192 304 32 24 \frac{256}{3} 288 \frac{1760}{3} -\frac{32}{3} \frac{18991}{15} \frac{236}{15} \frac{19204}{45} -\frac{319}{9} \frac{751}{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=4 is the signature of K11n133. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-1012345χ
15        1-1
13       211
11      21 -1
9     221 1
7    33   0
5   22    0
3  241    1
1 11      0
-1 2       2
-31        -1
Integral Khovanov Homology

(db, data source)

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

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

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K11n132.gif

K11n132

K11n134.gif

K11n134