K11a140

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K11a139.gif

K11a139

K11a141.gif

K11a141

Contents

K11a140.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a140 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X16,6,17,5 X18,8,19,7 X2,10,3,9 X22,11,1,12 X20,13,21,14 X8,16,9,15 X6,18,7,17 X14,19,15,20 X12,21,13,22
Gauss code 1, -5, 2, -1, 3, -9, 4, -8, 5, -2, 6, -11, 7, -10, 8, -3, 9, -4, 10, -7, 11, -6
Dowker-Thistlethwaite code 4 10 16 18 2 22 20 8 6 14 12
A Braid Representative
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A Morse Link Presentation K11a140 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a140/ThurstonBennequinNumber
Hyperbolic Volume 10.6275
A-Polynomial See Data:K11a140/A-polynomial

[edit Notes for K11a140's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -4

[edit Notes for K11a140's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+8 t^2-14 t+17-14 t^{-1} +8 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-4 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 65, 4 }
Jones polynomial -q^9+3 q^8-5 q^7+7 q^6-9 q^5+10 q^4-9 q^3+8 q^2-6 q+4-2 q^{-1} + q^{-2}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -z^6 a^{-4} -4 z^4 a^{-2} -3 z^4 a^{-4} +2 z^4 a^{-6} +z^4-5 z^2 a^{-2} -2 z^2 a^{-4} +5 z^2 a^{-6} -z^2 a^{-8} +3 z^2-2 a^{-2} +2 a^{-6} - a^{-8} +2
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +5 z^9 a^{-3} +3 z^9 a^{-5} -z^8 a^{-2} +3 z^8 a^{-4} +5 z^8 a^{-6} +z^8-11 z^7 a^{-1} -21 z^7 a^{-3} -4 z^7 a^{-5} +6 z^7 a^{-7} -13 z^6 a^{-2} -22 z^6 a^{-4} -9 z^6 a^{-6} +6 z^6 a^{-8} -6 z^6+19 z^5 a^{-1} +23 z^5 a^{-3} -10 z^5 a^{-5} -9 z^5 a^{-7} +5 z^5 a^{-9} +29 z^4 a^{-2} +24 z^4 a^{-4} -3 z^4 a^{-6} -7 z^4 a^{-8} +3 z^4 a^{-10} +12 z^4-11 z^3 a^{-1} -5 z^3 a^{-3} +11 z^3 a^{-5} -4 z^3 a^{-9} +z^3 a^{-11} -17 z^2 a^{-2} -5 z^2 a^{-4} +6 z^2 a^{-6} +2 z^2 a^{-8} -z^2 a^{-10} -9 z^2+z a^{-1} -2 z a^{-5} +z a^{-9} +2 a^{-2} -2 a^{-6} - a^{-8} +2
The A2 invariant Data:K11a140/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a140/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_25, 10_56,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a9,}

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 32 0 \frac{832}{3} -\frac{32}{3} 144 0 128 0 0 784 -\frac{1264}{3} \frac{2080}{3} \frac{448}{3} 64

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=4 is the signature of K11a140. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
19           1-1
17          2 2
15         31 -2
13        42  2
11       53   -2
9      54    1
7     45     1
5    45      -1
3   35       2
1  13        -2
-1 13         2
-3 1          -1
-51           1
Integral Khovanov Homology

(db, data source)

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

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K11a139.gif

K11a139

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K11a141