K11n30

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K11n29.gif

K11n29

K11n31.gif

K11n31

Contents

K11n30.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n30 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X5,13,6,12 X2837 X9,15,10,14 X11,18,12,19 X13,7,14,6 X15,1,16,22 X17,21,18,20 X19,10,20,11 X21,17,22,16
Gauss code 1, -4, 2, -1, -3, 7, 4, -2, -5, 10, -6, 3, -7, 5, -8, 11, -9, 6, -10, 9, -11, 8
Dowker-Thistlethwaite code 4 8 -12 2 -14 -18 -6 -22 -20 -10 -16
A Braid Representative
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A Morse Link Presentation K11n30 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n30/ThurstonBennequinNumber
Hyperbolic Volume 9.73657
A-Polynomial See Data:K11n30/A-polynomial

[edit Notes for K11n30'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 K11n30's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+6 t^2-6 t+5-6 t^{-1} +6 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-6 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 33, 4 }
Jones polynomial -q^9+3 q^8-4 q^7+5 q^6-6 q^5+5 q^4-4 q^3+3 q^2-q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} -z^6 a^{-6} +z^4 a^{-2} -4 z^4 a^{-4} -4 z^4 a^{-6} +z^4 a^{-8} +4 z^2 a^{-2} -4 z^2 a^{-4} -4 z^2 a^{-6} +4 z^2 a^{-8} +3 a^{-2} -2 a^{-4} -2 a^{-6} +3 a^{-8} - a^{-10}
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} +z^7 a^{-3} -4 z^7 a^{-5} -4 z^7 a^{-7} +z^7 a^{-9} +z^6 a^{-2} -2 z^6 a^{-4} -13 z^6 a^{-6} -10 z^6 a^{-8} -3 z^5 a^{-3} +8 z^5 a^{-5} +9 z^5 a^{-7} -2 z^5 a^{-9} -5 z^4 a^{-2} -3 z^4 a^{-4} +22 z^4 a^{-6} +23 z^4 a^{-8} +3 z^4 a^{-10} -11 z^3 a^{-5} -8 z^3 a^{-7} +4 z^3 a^{-9} +z^3 a^{-11} +7 z^2 a^{-2} +5 z^2 a^{-4} -15 z^2 a^{-6} -17 z^2 a^{-8} -4 z^2 a^{-10} +2 z a^{-3} +5 z a^{-5} +3 z a^{-7} -z a^{-9} -z a^{-11} -3 a^{-2} -2 a^{-4} +2 a^{-6} +3 a^{-8} + a^{-10}
The A2 invariant Data:K11n30/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n30/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: (0, 0)
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 0 0 80 48 0 480 160 160 0 0 0 0 1512 \frac{224}{3} 864 136 40

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 K11n30. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-101234567χ
19         1-1
17        2 2
15       21 -1
13      32  1
11     32   -1
9    23    -1
7   23     1
5  12      -1
3 13       2
1          0
-11         1
Integral Khovanov Homology

(db, data source)

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

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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K11n29.gif

K11n29

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K11n31