K11a367

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K11a366.gif

K11a366

K11n1.gif

K11n1

Contents

K11a367.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a367 at Knotilus!

K11a367 is the next knot in the sequence trefoil, cinquefoil, septafoil, nonafoil... (See also T(11,2).) K13a4878 comes after it.



Interlaced form of 11/2 star polygon or "undecagram"
Decorative interlaced form of 11/2 star polygon or "undecagram"
Decorative knotwork cross

Knot presentations

Planar diagram presentation X12,2,13,1 X14,4,15,3 X16,6,17,5 X18,8,19,7 X20,10,21,9 X22,12,1,11 X2,14,3,13 X4,16,5,15 X6,18,7,17 X8,20,9,19 X10,22,11,21
Gauss code 1, -7, 2, -8, 3, -9, 4, -10, 5, -11, 6, -1, 7, -2, 8, -3, 9, -4, 10, -5, 11, -6
Dowker-Thistlethwaite code 12 14 16 18 20 22 2 4 6 8 10
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gif
A Morse Link Presentation K11a367 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 5
3-genus 5
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a367/ThurstonBennequinNumber
Hyperbolic Volume Not hyperbolic
A-Polynomial See Data:K11a367/A-polynomial

[edit Notes for K11a367's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 5
Rasmussen s-Invariant -10

[edit Notes for K11a367's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^5-t^4+t^3-t^2+t-1+ t^{-1} - t^{-2} + t^{-3} - t^{-4} + t^{-5}
Conway polynomial z^{10}+9 z^8+28 z^6+35 z^4+15 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 11, 10 }
Jones polynomial -q^{16}+q^{15}-q^{14}+q^{13}-q^{12}+q^{11}-q^{10}+q^9-q^8+q^7+q^5
HOMFLY-PT polynomial (db, data sources) z^{10} a^{-10} +10 z^8 a^{-10} -z^8 a^{-12} +36 z^6 a^{-10} -8 z^6 a^{-12} +56 z^4 a^{-10} -21 z^4 a^{-12} +35 z^2 a^{-10} -20 z^2 a^{-12} +6 a^{-10} -5 a^{-12}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +z^9 a^{-11} +z^9 a^{-13} -10 z^8 a^{-10} -9 z^8 a^{-12} +z^8 a^{-14} -8 z^7 a^{-11} -7 z^7 a^{-13} +z^7 a^{-15} +36 z^6 a^{-10} +29 z^6 a^{-12} -6 z^6 a^{-14} +z^6 a^{-16} +21 z^5 a^{-11} +15 z^5 a^{-13} -5 z^5 a^{-15} +z^5 a^{-17} -56 z^4 a^{-10} -41 z^4 a^{-12} +10 z^4 a^{-14} -4 z^4 a^{-16} +z^4 a^{-18} -20 z^3 a^{-11} -10 z^3 a^{-13} +6 z^3 a^{-15} -3 z^3 a^{-17} +z^3 a^{-19} +35 z^2 a^{-10} +25 z^2 a^{-12} -4 z^2 a^{-14} +3 z^2 a^{-16} -2 z^2 a^{-18} +z^2 a^{-20} +5 z a^{-11} +z a^{-13} -z a^{-15} +z a^{-17} -z a^{-19} +z a^{-21} -6 a^{-10} -5 a^{-12}
The A2 invariant  q^{-18} + q^{-20} +2 q^{-22} + q^{-24} + q^{-26} - q^{-42} - q^{-44} - q^{-46}
The G2 invariant  q^{-90} + q^{-92} + q^{-94} + q^{-98} +2 q^{-100} +2 q^{-102} + q^{-104} + q^{-106} +2 q^{-108} +3 q^{-110} +2 q^{-112} + q^{-116} +2 q^{-118} + q^{-120} - q^{-124} + q^{-128} -2 q^{-132} - q^{-134} - q^{-138} - q^{-140} - q^{-142} - q^{-144} - q^{-150} - q^{-188} - q^{-190} - q^{-196} - q^{-198} - q^{-200} - q^{-206} - q^{-208} + q^{-264}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (15, 55)
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
60 440 1800 4230 610 26400 \frac{135344}{3} \frac{23936}{3} 5368 36000 96800 253800 36600 \frac{985183}{2} \frac{70174}{3} 176338 \frac{4919}{2} \frac{44287}{2}

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=10 is the signature of K11a367. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
33           1-1
31            0
29         11 0
27            0
25       11   0
23            0
21     11     0
19            0
17   11       0
15            0
13  1         1
111           1
91           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=9 i=11
r=0 {\mathbb Z} {\mathbb Z}
r=1
r=2 {\mathbb Z}
r=3 {\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}
r=5 {\mathbb Z}_2 {\mathbb Z}
r=6 {\mathbb Z}
r=7 {\mathbb Z}_2 {\mathbb Z}
r=8 {\mathbb Z}
r=9 {\mathbb Z}_2 {\mathbb Z}
r=10 {\mathbb Z}
r=11 {\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

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K11a366.gif

K11a366

K11n1.gif

K11n1