K11n139

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K11n138.gif

K11n138

K11n140.gif

K11n140

Contents

K11n139.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n139 at Knotilus!

K11n139 is also known as the pretzel knot P(5,3,-3).



Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X16,6,17,5 X14,8,15,7 X9,21,10,20 X11,19,12,18 X2,13,3,14 X6,16,7,15 X17,22,18,1 X19,11,20,10 X21,9,22,8
Gauss code 1, -7, 2, -1, 3, -8, 4, 11, -5, 10, -6, -2, 7, -4, 8, -3, -9, 6, -10, 5, -11, 9
Dowker-Thistlethwaite code 4 12 16 14 -20 -18 2 6 -22 -10 -8
A Braid Representative
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A Morse Link Presentation K11n139 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 1
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n139/ThurstonBennequinNumber
Hyperbolic Volume 5.56972
A-Polynomial See Data:K11n139/A-polynomial

[edit Notes for K11n139's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n139's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t+5-2 t^{-1}
Conway polynomial 1-2 z^2
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 9, 0 }
Jones polynomial q^8-q^7+q^6-2 q^5+q^4-q^3+q^2+1
HOMFLY-PT polynomial (db, data sources) -z^2 a^{-4} -z^2 a^{-6} + a^{-2} - a^{-4} - a^{-6} + a^{-8} +1
Kauffman polynomial (db, data sources) z^9 a^{-5} +z^9 a^{-7} +z^8 a^{-4} +2 z^8 a^{-6} +z^8 a^{-8} -7 z^7 a^{-5} -7 z^7 a^{-7} -6 z^6 a^{-4} -13 z^6 a^{-6} -7 z^6 a^{-8} +z^5 a^{-3} +17 z^5 a^{-5} +16 z^5 a^{-7} +10 z^4 a^{-4} +25 z^4 a^{-6} +15 z^4 a^{-8} -5 z^3 a^{-3} -20 z^3 a^{-5} -15 z^3 a^{-7} -5 z^2 a^{-4} -15 z^2 a^{-6} -10 z^2 a^{-8} +4 z a^{-3} +10 z a^{-5} +6 z a^{-7} - a^{-2} - a^{-4} + a^{-6} + a^{-8} +1
The A2 invariant Data:K11n139/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n139/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {6_1, 9_46, K11n67, K11n97,}

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

Vassiliev invariants

V2 and V3: (-2, -5)
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 -40 32 -\frac{172}{3} \frac{52}{3} 320 \frac{1040}{3} \frac{320}{3} 120 -\frac{256}{3} 800 \frac{1376}{3} -\frac{416}{3} \frac{34889}{15} -\frac{1036}{15} \frac{46076}{45} \frac{1639}{9} \frac{1289}{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=0 is the signature of K11n139. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
012345678χ
17        11
15         0
13      11 0
11     1   -1
9     1   -1
7   11    0
5         0
3  1      1
11        1
-11        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=0 {\mathbb Z} {\mathbb Z}
r=1
r=2 {\mathbb Z}
r=3 {\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=6 {\mathbb Z}_2 {\mathbb Z}
r=7 {\mathbb Z}
r=8 {\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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K11n138.gif

K11n138

K11n140.gif

K11n140