K11n104

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K11n103.gif

K11n103

K11n105.gif

K11n105

Contents

K11n104.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n104 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 X15,19,16,18 X17,6,18,7 X19,9,20,8 X21,13,22,12
Gauss code 1, -5, 2, -1, -3, 9, -4, 10, 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 -6 -8 -12
A Braid Representative
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A Morse Link Presentation K11n104 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n104/ThurstonBennequinNumber
Hyperbolic Volume 7.34942
A-Polynomial See Data:K11n104/A-polynomial

[edit Notes for K11n104's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant -6

[edit Notes for K11n104's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+3 t^3-2 t^2-2 t+5-2 t^{-1} -2 t^{-2} +3 t^{-3} - t^{-4}
Conway polynomial -z^8-5 z^6-4 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 3, 6 }
Jones polynomial q^{10}-q^9-q^7+q^5-q^4+2 q^3-q^2+q
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-6} +z^6 a^{-4} -7 z^6 a^{-6} +z^6 a^{-8} +6 z^4 a^{-4} -16 z^4 a^{-6} +6 z^4 a^{-8} +10 z^2 a^{-4} -16 z^2 a^{-6} +8 z^2 a^{-8} -z^2 a^{-10} +5 a^{-4} -6 a^{-6} +2 a^{-8}
Kauffman polynomial (db, data sources) z^9 a^{-5} +z^9 a^{-7} +z^8 a^{-4} +3 z^8 a^{-6} +2 z^8 a^{-8} -6 z^7 a^{-5} -5 z^7 a^{-7} +z^7 a^{-9} -7 z^6 a^{-4} -20 z^6 a^{-6} -13 z^6 a^{-8} +9 z^5 a^{-5} +2 z^5 a^{-7} -6 z^5 a^{-9} +z^5 a^{-11} +16 z^4 a^{-4} +39 z^4 a^{-6} +22 z^4 a^{-8} +z^4 a^{-12} -z^3 a^{-5} +9 z^3 a^{-7} +6 z^3 a^{-9} -4 z^3 a^{-11} -15 z^2 a^{-4} -27 z^2 a^{-6} -11 z^2 a^{-8} -2 z^2 a^{-10} -3 z^2 a^{-12} -4 z a^{-5} -5 z a^{-7} +z a^{-9} +2 z a^{-11} +5 a^{-4} +6 a^{-6} +2 a^{-8}
The A2 invariant  q^{-4} + q^{-6} + q^{-8} +2 q^{-10} + q^{-12} -2 q^{-20} - q^{-22} -2 q^{-24} + q^{-30} +2 q^{-32} - q^{-34}
The G2 invariant Data:K11n104/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: (1, -1)
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
4 -8 8 -\frac{82}{3} \frac{106}{3} -32 \frac{112}{3} \frac{256}{3} 184 \frac{32}{3} 32 -\frac{328}{3} \frac{424}{3} \frac{19711}{30} -\frac{8422}{15} \frac{72422}{45} \frac{3713}{18} \frac{4831}{30}

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=6 is the signature of K11n104. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-1012345678χ
21          11
19           0
17       111 -1
15      12   -1
13     111   -1
11    122    1
9   11      0
7  111      1
5 12        1
3           0
11          1
Integral Khovanov Homology

(db, data source)

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

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K11n103.gif

K11n103

K11n105.gif

K11n105