K11a1

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10 165.gif

10_165

K11a2.gif

K11a2

K11a1.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a1 at Knotilus!


K11a1.
A graph, Knot K11a1.

Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X14,7,15,8 X2,9,3,10 X16,12,17,11 X20,14,21,13 X6,15,7,16 X22,18,1,17 X12,20,13,19 X18,22,19,21
Gauss code 1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -10, 7, -4, 8, -6, 9, -11, 10, -7, 11, -9
Dowker-Thistlethwaite code 4 8 10 14 2 16 20 6 22 12 18
A Braid Representative
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A Morse Link Presentation K11a1 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a1/ThurstonBennequinNumber
Hyperbolic Volume 15.6439
A-Polynomial See Data:K11a1/A-polynomial

[edit Notes for K11a1's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a1's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 127, 2 }
Jones polynomial
HOMFLY-PT polynomial (db, data sources)
Kauffman polynomial (db, data sources)
The A2 invariant Data:K11a1/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a1/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a122, K11a149,}

Same Jones Polynomial (up to mirroring, ): {K11a149,}

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

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 are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where 2 is the signature of K11a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
17           1-1
15          3 3
13         61 -5
11        83  5
9       106   -4
7      118    3
5     910     1
3    811      -3
1   510       5
-1  27        -5
-3 15         4
-5 2          -2
-71           1
Integral Khovanov Homology

(db, data source)

  

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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10 165.gif

10_165

K11a2.gif

K11a2