K11n38

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K11n37.gif

K11n37

K11n39.gif

K11n39

Contents

K11n38.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n38 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X5,12,6,13 X2837 X9,19,10,18 X11,6,12,7 X13,22,14,1 X15,20,16,21 X17,11,18,10 X19,16,20,17 X21,14,22,15
Gauss code 1, -4, 2, -1, -3, 6, 4, -2, -5, 9, -6, 3, -7, 11, -8, 10, -9, 5, -10, 8, -11, 7
Dowker-Thistlethwaite code 4 8 -12 2 -18 -6 -22 -20 -10 -16 -14
A Braid Representative
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A Morse Link Presentation K11n38 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n38/ThurstonBennequinNumber
Hyperbolic Volume 4.1249
A-Polynomial See Data:K11n38/A-polynomial

[edit Notes for K11n38'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 K11n38's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^2+t+1+ t^{-1} - t^{-2}
Conway polynomial -z^4-3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 3, 2 }
Jones polynomial q^2- q^{-2} + q^{-3} - q^{-4} + q^{-5}
HOMFLY-PT polynomial (db, data sources) z^2 a^4+2 a^4-z^4 a^2-4 z^2 a^2-3 a^2+1+ a^{-2}
Kauffman polynomial (db, data sources) a^3 z^9+a z^9+a^4 z^8+2 a^2 z^8+z^8-7 a^3 z^7-7 a z^7-7 a^4 z^6-14 a^2 z^6-7 z^6+15 a^3 z^5+15 a z^5+15 a^4 z^4+29 a^2 z^4+14 z^4-12 a^3 z^3-13 a z^3-z^3 a^{-1} -11 a^4 z^2-20 a^2 z^2-9 z^2+4 a^3 z+5 a z+z a^{-1} +2 a^4+3 a^2- a^{-2} +1
The A2 invariant Data:K11n38/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n38/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n102,}

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 98 46 -192 -\frac{992}{3} -\frac{224}{3} -80 -288 128 -1176 -552 -\frac{7311}{10} \frac{942}{5} -\frac{4474}{5} \frac{389}{2} -\frac{2191}{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=2 is the signature of K11n38. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012χ
5        11
3      1  1
1      11 0
-1    121  0
-3   1     -1
-5   11    0
-7 11      0
-9         0
-111        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1 i=3
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z} {\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}_2 {\mathbb Z}^{2}
r=0 {\mathbb Z} {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=1 {\mathbb Z}
r=2 {\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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K11n37.gif

K11n37

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K11n39