K11n140

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K11n139.gif

K11n139

K11n141.gif

K11n141

Contents

K11n140.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n140 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X5,16,6,17 X7,14,8,15 X20,9,21,10 X18,11,19,12 X2,13,3,14 X15,6,16,7 X22,18,1,17 X10,19,11,20 X8,21,9,22
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 K11n140 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:K11n140/ThurstonBennequinNumber
Hyperbolic Volume 11.4116
A-Polynomial See Data:K11n140/A-polynomial

[edit Notes for K11n140's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n140's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^2+13 t-21+13 t^{-1} -2 t^{-2}
Conway polynomial -2 z^4+5 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 51, -2 }
Jones polynomial q-3+5 q^{-1} -7 q^{-2} +9 q^{-3} -8 q^{-4} +8 q^{-5} -5 q^{-6} +3 q^{-7} -2 q^{-8}
HOMFLY-PT polynomial (db, data sources) -2 a^8+3 z^2 a^6+2 a^6-z^4 a^4+z^2 a^4+a^4-z^4 a^2+z^2
Kauffman polynomial (db, data sources) 3 z^5 a^9-10 z^3 a^9+7 z a^9+z^8 a^8-2 z^6 a^8+z^2 a^8-2 a^8+z^9 a^7-3 z^7 a^7+9 z^5 a^7-17 z^3 a^7+9 z a^7+4 z^8 a^6-12 z^6 a^6+15 z^4 a^6-6 z^2 a^6-2 a^6+z^9 a^5+z^7 a^5-3 z^5 a^5+z^3 a^5+2 z a^5+3 z^8 a^4-6 z^6 a^4+9 z^4 a^4-5 z^2 a^4+a^4+4 z^7 a^3-6 z^5 a^3+4 z^3 a^3+4 z^6 a^2-5 z^4 a^2+z^2 a^2+3 z^5 a-4 z^3 a+z^4-z^2
The A2 invariant Data:K11n140/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n140/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (5, -11)
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
20 -88 200 \frac{1606}{3} \frac{338}{3} -1760 -\frac{10288}{3} -\frac{1696}{3} -696 \frac{4000}{3} 3872 \frac{32120}{3} \frac{6760}{3} \frac{131935}{6} -\frac{5030}{3} \frac{106310}{9} \frac{4837}{18} \frac{11071}{6}

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 K11n140. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-7-6-5-4-3-2-1012χ
3         11
1        2 -2
-1       31 2
-3      53  -2
-5     42   2
-7    45    1
-9   44     0
-11  14      3
-13 24       -2
-15 1        1
-172         -2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-7 {\mathbb Z}^{2}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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K11n139

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K11n141