K11n29

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K11n28.gif

K11n28

K11n30.gif

K11n30

Contents

K11n29.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n29 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X5,13,6,12 X2837 X14,9,15,10 X18,12,19,11 X13,7,14,6 X22,15,1,16 X20,17,21,18 X10,20,11,19 X16,21,17,22
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 K11n29 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n29/ThurstonBennequinNumber
Hyperbolic Volume 11.7974
A-Polynomial See Data:K11n29/A-polynomial

[edit Notes for K11n29's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n29's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^2+13 t-19+13 t^{-1} -3 t^{-2}
Conway polynomial -3 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 51, 2 }
Jones polynomial -q^8+2 q^7-4 q^6+7 q^5-8 q^4+9 q^3-8 q^2+6 q-4+2 q^{-1}
HOMFLY-PT polynomial (db, data sources) -2 z^4 a^{-2} -z^4 a^{-4} -4 z^2 a^{-2} +z^2 a^{-4} +2 z^2 a^{-6} +2 z^2-3 a^{-2} +2 a^{-4} + a^{-6} - a^{-8} +2
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} -z^7 a^{-3} +2 z^7 a^{-7} -6 z^6 a^{-2} -11 z^6 a^{-4} -3 z^6 a^{-6} +2 z^6 a^{-8} +z^5 a^{-1} +4 z^5 a^{-3} -2 z^5 a^{-7} +z^5 a^{-9} +14 z^4 a^{-2} +18 z^4 a^{-4} +2 z^4 a^{-6} -5 z^4 a^{-8} +3 z^4-5 z^3 a^{-1} -5 z^3 a^{-3} -3 z^3 a^{-7} -3 z^3 a^{-9} -14 z^2 a^{-2} -11 z^2 a^{-4} +3 z^2 a^{-8} -6 z^2+2 z a^{-1} +3 z a^{-3} +z a^{-5} +2 z a^{-7} +2 z a^{-9} +3 a^{-2} +2 a^{-4} - a^{-6} - a^{-8} +2
The A2 invariant Data:K11n29/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n29/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_36, K11a230,}

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

Vassiliev invariants

V2 and V3: (1, 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
4 32 8 \frac{398}{3} \frac{82}{3} 128 \frac{1472}{3} \frac{224}{3} 96 \frac{32}{3} 512 \frac{1592}{3} \frac{328}{3} \frac{66751}{30} -\frac{474}{5} \frac{48062}{45} \frac{737}{18} \frac{3871}{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=2 is the signature of K11n29. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-101234567χ
17         1-1
15        1 1
13       31 -2
11      41  3
9     43   -1
7    54    1
5   34     1
3  35      -2
1 24       2
-1 2        -2
-32         2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-2 {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11n28.gif

K11n28

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K11n30