K11n60

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K11n59.gif

K11n59

K11n61.gif

K11n61

Contents

K11n60.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n60 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X5,15,6,14 X2837 X9,17,10,16 X11,18,12,19 X13,20,14,21 X15,7,16,6 X17,1,18,22 X19,12,20,13 X21,10,22,11
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.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation K11n60 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n60/ThurstonBennequinNumber
Hyperbolic Volume 9.48716
A-Polynomial See Data:K11n60/A-polynomial

[edit Notes for K11n60's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n60's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+3 t^3-4 t^2+5 t-5+5 t^{-1} -4 t^{-2} +3 t^{-3} - t^{-4}
Conway polynomial -z^8-5 z^6-6 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 31, 2 }
Jones polynomial -q^6+2 q^5-3 q^4+5 q^3-5 q^2+5 q-4+3 q^{-1} -2 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-2} -7 z^6 a^{-2} +z^6 a^{-4} +z^6-17 z^4 a^{-2} +6 z^4 a^{-4} +5 z^4-17 z^2 a^{-2} +11 z^2 a^{-4} -z^2 a^{-6} +7 z^2-6 a^{-2} +6 a^{-4} -2 a^{-6} +3
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +4 z^8 a^{-2} +2 z^8 a^{-4} +2 z^8+2 a z^7-2 z^7 a^{-1} -3 z^7 a^{-3} +z^7 a^{-5} +a^2 z^6-20 z^6 a^{-2} -11 z^6 a^{-4} -8 z^6-8 a z^5-4 z^5 a^{-1} -4 z^5 a^{-5} -4 a^2 z^4+35 z^4 a^{-2} +24 z^4 a^{-4} +2 z^4 a^{-6} +9 z^4+7 a z^3+8 z^3 a^{-1} +8 z^3 a^{-3} +8 z^3 a^{-5} +z^3 a^{-7} +3 a^2 z^2-25 z^2 a^{-2} -19 z^2 a^{-4} -4 z^2 a^{-6} -7 z^2-2 a z-4 z a^{-1} -4 z a^{-3} -4 z a^{-5} -2 z a^{-7} +6 a^{-2} +6 a^{-4} +2 a^{-6} +3
The A2 invariant q^8+q^4-2 q^{-4} + q^{-6} - q^{-8} +2 q^{-10} + q^{-12} + q^{-14} + q^{-16} - q^{-18} - q^{-22}
The G2 invariant Data:K11n60/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_46,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {9_8,}

Vassiliev invariants

V2 and V3: (0, 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
0 16 0 96 48 0 \frac{544}{3} \frac{160}{3} 48 0 128 0 0 464 240 -112 112 -80

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=2 is the signature of K11n60. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-1012345χ
13         1-1
11        1 1
9       21 -1
7      31  2
5     22   0
3    33    0
1   23     1
-1  12      -1
-3 12       1
-5 1        -1
-71         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\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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See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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K11n59

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K11n61