K11n107

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K11n106.gif

K11n106

K11n108.gif

K11n108

Contents

K11n107.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n107 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X5,15,6,14 X7,16,8,17 X2,10,3,9 X11,21,12,20 X13,1,14,22 X15,18,16,19 X17,8,18,9 X19,7,20,6 X21,13,22,12
Gauss code 1, -5, 2, -1, -3, 10, -4, 9, 5, -2, -6, 11, -7, 3, -8, 4, -9, 8, -10, 6, -11, 7
Dowker-Thistlethwaite code 4 10 -14 -16 2 -20 -22 -18 -8 -6 -12
A Braid Representative
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A Morse Link Presentation K11n107 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n107/ThurstonBennequinNumber
Hyperbolic Volume 9.27292
A-Polynomial See Data:K11n107/A-polynomial

[edit Notes for K11n107's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n107's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-3 t^3+4 t^2-2 t+1-2 t^{-1} +4 t^{-2} -3 t^{-3} + t^{-4}
Conway polynomial z^8+5 z^6+6 z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 21, 4 }
Jones polynomial -q^7+2 q^6-3 q^5+3 q^4-3 q^3+4 q^2-2 q+2- q^{-1}
HOMFLY-PT polynomial (db, data sources) z^8 a^{-4} -z^6 a^{-2} +7 z^6 a^{-4} -z^6 a^{-6} -5 z^4 a^{-2} +17 z^4 a^{-4} -6 z^4 a^{-6} -6 z^2 a^{-2} +18 z^2 a^{-4} -10 z^2 a^{-6} +z^2 a^{-8} - a^{-2} +6 a^{-4} -5 a^{-6} + a^{-8}
Kauffman polynomial (db, data sources) z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +4 z^8 a^{-4} +2 z^8 a^{-6} +z^7 a^{-1} -3 z^7 a^{-3} -3 z^7 a^{-5} +z^7 a^{-7} -11 z^6 a^{-2} -22 z^6 a^{-4} -11 z^6 a^{-6} -5 z^5 a^{-1} -3 z^5 a^{-3} -3 z^5 a^{-5} -5 z^5 a^{-7} +18 z^4 a^{-2} +37 z^4 a^{-4} +19 z^4 a^{-6} +6 z^3 a^{-1} +8 z^3 a^{-3} +9 z^3 a^{-5} +7 z^3 a^{-7} -11 z^2 a^{-2} -26 z^2 a^{-4} -15 z^2 a^{-6} -z a^{-1} -2 z a^{-3} -4 z a^{-5} -3 z a^{-7} + a^{-2} +6 a^{-4} +5 a^{-6} + a^{-8}
The A2 invariant -q^2+ q^{-4} +2 q^{-6} +2 q^{-8} +2 q^{-10} - q^{-12} + q^{-14} - q^{-16} - q^{-20} - q^{-22} - q^{-26} + q^{-28}
The G2 invariant Data:K11n107/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: (3, 4)
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 32 72 94 -14 384 \frac{1088}{3} \frac{128}{3} -64 288 512 1128 -168 \frac{14991}{10} \frac{9574}{15} -\frac{8338}{15} -\frac{175}{6} -\frac{1489}{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=4 is the signature of K11n107. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-1012345χ
15        1-1
13       1 1
11      21 -1
9     11  0
7    22   0
5   21    1
3  13     2
1 11      0
-1 1       1
-31        -1
Integral Khovanov Homology

(db, data source)

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

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K11n106.gif

K11n106

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K11n108