K11n52

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K11n51.gif

K11n51

K11n53.gif

K11n53

Contents

K11n52.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n52 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X5,14,6,15 X2837 X16,10,17,9 X18,11,19,12 X13,6,14,7 X22,16,1,15 X20,18,21,17 X10,19,11,20 X12,22,13,21
Gauss code 1, -4, 2, -1, -3, 7, 4, -2, 5, -10, 6, -11, -7, 3, 8, -5, 9, -6, 10, -9, 11, -8
Dowker-Thistlethwaite code 4 8 -14 2 16 18 -6 22 20 10 12
A Braid Representative
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A Morse Link Presentation K11n52 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n52/ThurstonBennequinNumber
Hyperbolic Volume 12.5877
A-Polynomial See Data:K11n52/A-polynomial

[edit Notes for K11n52's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -2

[edit Notes for K11n52's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-6 t^2+14 t-17+14 t^{-1} -6 t^{-2} + t^{-3}
Conway polynomial z^6-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 59, 2 }
Jones polynomial -q^8+3 q^7-5 q^6+8 q^5-10 q^4+10 q^3-9 q^2+7 q-4+2 q^{-1}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-4} -3 z^4 a^{-2} +4 z^4 a^{-4} -z^4 a^{-6} -8 z^2 a^{-2} +7 z^2 a^{-4} -2 z^2 a^{-6} +2 z^2-5 a^{-2} +4 a^{-4} - a^{-6} +3
Kauffman polynomial (db, data sources) z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +5 z^8 a^{-4} +3 z^8 a^{-6} +z^7 a^{-1} +z^7 a^{-3} +4 z^7 a^{-5} +4 z^7 a^{-7} -5 z^6 a^{-2} -12 z^6 a^{-4} -4 z^6 a^{-6} +3 z^6 a^{-8} +z^5 a^{-1} -2 z^5 a^{-3} -13 z^5 a^{-5} -9 z^5 a^{-7} +z^5 a^{-9} +14 z^4 a^{-2} +18 z^4 a^{-4} -7 z^4 a^{-8} +3 z^4-3 z^3 a^{-1} +4 z^3 a^{-3} +15 z^3 a^{-5} +6 z^3 a^{-7} -2 z^3 a^{-9} -17 z^2 a^{-2} -13 z^2 a^{-4} +z^2 a^{-6} +3 z^2 a^{-8} -6 z^2-3 z a^{-3} -5 z a^{-5} -2 z a^{-7} +5 a^{-2} +4 a^{-4} + a^{-6} +3
The A2 invariant Data:K11n52/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n52/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_32, K11n124,}

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

Vassiliev invariants

V2 and V3: (-1, 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
-4 0 8 \frac{82}{3} \frac{14}{3} 0 0 -32 0 -\frac{32}{3} 0 -\frac{328}{3} -\frac{56}{3} -\frac{6751}{30} -\frac{806}{5} -\frac{1022}{45} -\frac{65}{18} \frac{449}{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 K11n52. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-101234567χ
17         1-1
15        2 2
13       31 -2
11      52  3
9     53   -2
7    55    0
5   45     1
3  35      -2
1 25       3
-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}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11n51.gif

K11n51

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K11n53