K11a141

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K11a140.gif

K11a140

K11a142.gif

K11a142

Contents

K11a141.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a141 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X16,6,17,5 X18,7,19,8 X12,10,13,9 X2,11,3,12 X8,14,9,13 X20,15,21,16 X22,18,1,17 X14,19,15,20 X6,21,7,22
Gauss code 1, -6, 2, -1, 3, -11, 4, -7, 5, -2, 6, -5, 7, -10, 8, -3, 9, -4, 10, -8, 11, -9
Dowker-Thistlethwaite code 4 10 16 18 12 2 8 20 22 14 6
A Braid Representative
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A Morse Link Presentation K11a141 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a141/ThurstonBennequinNumber
Hyperbolic Volume 14.9456
A-Polynomial See Data:K11a141/A-polynomial

[edit Notes for K11a141's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 2

[edit Notes for K11a141's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-11 t^2+24 t-29+24 t^{-1} -11 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 103, -2 }
Jones polynomial -q^4+4 q^3-7 q^2+11 q-14+16 q^{-1} -16 q^{-2} +14 q^{-3} -10 q^{-4} +6 q^{-5} -3 q^{-6} + q^{-7}
HOMFLY-PT polynomial (db, data sources) z^2 a^6+a^6-2 z^4 a^4-4 z^2 a^4-2 a^4+z^6 a^2+2 z^4 a^2+2 z^2 a^2+2 a^2+z^6+2 z^4-1-z^4 a^{-2} -z^2 a^{-2} + a^{-2}
Kauffman polynomial (db, data sources) 2 a^2 z^{10}+2 z^{10}+5 a^3 z^9+10 a z^9+5 z^9 a^{-1} +6 a^4 z^8+5 a^2 z^8+4 z^8 a^{-2} +3 z^8+6 a^5 z^7-6 a^3 z^7-30 a z^7-17 z^7 a^{-1} +z^7 a^{-3} +5 a^6 z^6-5 a^4 z^6-23 a^2 z^6-15 z^6 a^{-2} -28 z^6+3 a^7 z^5-5 a^5 z^5-a^3 z^5+24 a z^5+14 z^5 a^{-1} -3 z^5 a^{-3} +a^8 z^4-5 a^6 z^4-5 a^4 z^4+16 a^2 z^4+15 z^4 a^{-2} +30 z^4-3 a^7 z^3-7 a z^3-2 z^3 a^{-1} +2 z^3 a^{-3} -a^8 z^2+3 a^6 z^2+7 a^4 z^2-3 z^2 a^{-2} -6 z^2+a^7 z+a^5 z+a^3 z+a z-a^6-2 a^4-2 a^2- a^{-2} -1
The A2 invariant Data:K11a141/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a141/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a12,}

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

Vassiliev invariants

V2 and V3: (-2, 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
-8 16 32 \frac{20}{3} \frac{28}{3} -128 -\frac{512}{3} -\frac{224}{3} -16 -\frac{256}{3} 128 -\frac{160}{3} -\frac{224}{3} \frac{5849}{15} -\frac{252}{5} \frac{11636}{45} \frac{7}{9} \frac{569}{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 K11a141. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012345χ
9           1-1
7          3 3
5         41 -3
3        73  4
1       74   -3
-1      97    2
-3     88     0
-5    68      -2
-7   48       4
-9  26        -4
-11 14         3
-13 2          -2
-151           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\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.

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K11a140.gif

K11a140

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K11a142