K11n18

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K11n17.gif

K11n17

K11n19.gif

K11n19

Contents

K11n18.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n18 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X7,16,8,17 X2,9,3,10 X11,19,12,18 X13,21,14,20 X15,6,16,7 X17,1,18,22 X19,15,20,14 X21,13,22,12
Gauss code 1, -5, 2, -1, 3, 8, -4, -2, 5, -3, -6, 11, -7, 10, -8, 4, -9, 6, -10, 7, -11, 9
Dowker-Thistlethwaite code 4 8 10 -16 2 -18 -20 -6 -22 -14 -12
A Braid Representative
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A Morse Link Presentation K11n18 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n18/ThurstonBennequinNumber
Hyperbolic Volume 10.1473
A-Polynomial See Data:K11n18/A-polynomial

[edit Notes for K11n18's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 2
Rasmussen s-Invariant 0

[edit Notes for K11n18's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^2-8 t+13-8 t^{-1} +2 t^{-2}
Conway polynomial 2 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 33, 0 }
Jones polynomial -q^7+2 q^6-3 q^5+5 q^4-5 q^3+5 q^2-5 q+4-2 q^{-1} + q^{-2}
HOMFLY-PT polynomial (db, data sources) z^4 a^{-2} +z^4 a^{-4} +z^2 a^{-2} +2 z^2 a^{-4} -z^2 a^{-6} -2 z^2+a^2+2 a^{-4} - a^{-6} -1
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} +2 z^7 a^{-1} -z^7 a^{-3} -2 z^7 a^{-5} +z^7 a^{-7} -6 z^6 a^{-2} -17 z^6 a^{-4} -10 z^6 a^{-6} +z^6-6 z^5 a^{-1} -6 z^5 a^{-3} -5 z^5 a^{-5} -5 z^5 a^{-7} +4 z^4 a^{-2} +21 z^4 a^{-4} +15 z^4 a^{-6} -2 z^4+2 a z^3+6 z^3 a^{-1} +6 z^3 a^{-3} +9 z^3 a^{-5} +7 z^3 a^{-7} +a^2 z^2-z^2 a^{-2} -12 z^2 a^{-4} -8 z^2 a^{-6} +4 z^2-a z-z a^{-1} -z a^{-3} -3 z a^{-5} -2 z a^{-7} -a^2+2 a^{-4} + a^{-6} -1
The A2 invariant Data:K11n18/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n18/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_146, K11n62,}

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

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 16 -16 0 -\frac{32}{3} -\frac{128}{3} -16 0 128 0 0 168 -16 \frac{64}{3} -\frac{56}{3} -8

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=0 is the signature of K11n18. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-101234567χ
15         1-1
13        1 1
11       21 -1
9      31  2
7     22   0
5    33    0
3   22     0
1  23      -1
-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=-1 i=1
r=-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}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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K11n17.gif

K11n17

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K11n19