K11n79

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K11n78.gif

K11n78

K11n80.gif

K11n80

Contents

K11n79.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n79 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X5,14,6,15 X2837 X9,21,10,20 X11,19,12,18 X13,6,14,7 X15,22,16,1 X17,13,18,12 X19,11,20,10 X21,16,22,17
Gauss code 1, -4, 2, -1, -3, 7, 4, -2, -5, 10, -6, 9, -7, 3, -8, 11, -9, 6, -10, 5, -11, 8
Dowker-Thistlethwaite code 4 8 -14 2 -20 -18 -6 -22 -12 -10 -16
A Braid Representative
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A Morse Link Presentation K11n79 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n79/ThurstonBennequinNumber
Hyperbolic Volume 6.75197
A-Polynomial See Data:K11n79/A-polynomial

[edit Notes for K11n79's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 2
Rasmussen s-Invariant 0

[edit Notes for K11n79's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^2+4 t-3+4 t^{-1} -2 t^{-2}
Conway polynomial -2 z^4-4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 15, 2 }
Jones polynomial q^5-q^4+2 q^3-3 q^2+2 q-2+2 q^{-1} - q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) -z^4 a^{-2} -z^4+a^2 z^2-3 z^2 a^{-2} +z^2 a^{-4} -3 z^2+2 a^2-2 a^{-2} +2 a^{-4} -1
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +2 z^8 a^{-2} +z^8 a^{-4} +z^8+a z^7-6 z^7 a^{-1} -7 z^7 a^{-3} +a^2 z^6-12 z^6 a^{-2} -7 z^6 a^{-4} -4 z^6-4 a z^5+13 z^5 a^{-1} +17 z^5 a^{-3} -5 a^2 z^4+23 z^4 a^{-2} +16 z^4 a^{-4} +2 z^4+2 a z^3-15 z^3 a^{-1} -17 z^3 a^{-3} +6 a^2 z^2-16 z^2 a^{-2} -12 z^2 a^{-4} +2 z^2+a z+7 z a^{-1} +7 z a^{-3} +z a^{-5} -2 a^2+2 a^{-2} +2 a^{-4} -1
The A2 invariant Data:K11n79/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n79/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n138,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n138,}

Vassiliev invariants

V2 and V3: (-4, -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
-16 -16 128 \frac{568}{3} \frac{248}{3} 256 \frac{1280}{3} \frac{320}{3} 80 -\frac{2048}{3} 128 -\frac{9088}{3} -\frac{3968}{3} -\frac{37862}{15} \frac{3336}{5} -\frac{119528}{45} \frac{3350}{9} -\frac{9302}{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=2 is the signature of K11n79. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234χ
11        11
9         0
7      21 1
5     1   -1
3    12   -1
1   22    0
-1   11    0
-3 12      1
-5         0
-71        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1 i=3
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{2}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z} {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z}
r=4 {\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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K11n78

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K11n80