K11a274

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K11a273.gif

K11a273

K11a275.gif

K11a275

Contents

K11a274.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a274 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,4,11,3 X14,5,15,6 X18,7,19,8 X2,10,3,9 X20,11,21,12 X22,14,1,13 X4,15,5,16 X12,17,13,18 X8,19,9,20 X16,22,17,21
Gauss code 1, -5, 2, -8, 3, -1, 4, -10, 5, -2, 6, -9, 7, -3, 8, -11, 9, -4, 10, -6, 11, -7
Dowker-Thistlethwaite code 6 10 14 18 2 20 22 4 12 8 16
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation K11a274 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a274/ThurstonBennequinNumber
Hyperbolic Volume 18.6505
A-Polynomial See Data:K11a274/A-polynomial

[edit Notes for K11a274's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant 0

[edit Notes for K11a274's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-6 t^3+18 t^2-35 t+45-35 t^{-1} +18 t^{-2} -6 t^{-3} + t^{-4}
Conway polynomial z^8+2 z^6+2 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 165, 0 }
Jones polynomial -q^5+4 q^4-10 q^3+18 q^2-23 q+27-27 q^{-1} +23 q^{-2} -17 q^{-3} +10 q^{-4} -4 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) z^8-2 a^2 z^6-z^6 a^{-2} +5 z^6+a^4 z^4-7 a^2 z^4-3 z^4 a^{-2} +11 z^4+2 a^4 z^2-10 a^2 z^2-4 z^2 a^{-2} +11 z^2+2 a^4-5 a^2- a^{-2} +5
Kauffman polynomial (db, data sources) 3 a^2 z^{10}+3 z^{10}+8 a^3 z^9+18 a z^9+10 z^9 a^{-1} +8 a^4 z^8+16 a^2 z^8+13 z^8 a^{-2} +21 z^8+4 a^5 z^7-10 a^3 z^7-30 a z^7-7 z^7 a^{-1} +9 z^7 a^{-3} +a^6 z^6-17 a^4 z^6-51 a^2 z^6-22 z^6 a^{-2} +4 z^6 a^{-4} -59 z^6-8 a^5 z^5-3 a^3 z^5+10 a z^5-8 z^5 a^{-1} -12 z^5 a^{-3} +z^5 a^{-5} -2 a^6 z^4+13 a^4 z^4+48 a^2 z^4+19 z^4 a^{-2} -4 z^4 a^{-4} +56 z^4+5 a^5 z^3+4 a^3 z^3-a z^3+8 z^3 a^{-1} +7 z^3 a^{-3} -z^3 a^{-5} +a^6 z^2-6 a^4 z^2-24 a^2 z^2-9 z^2 a^{-2} +z^2 a^{-4} -27 z^2-a^5 z+a z-z a^{-1} -z a^{-3} +2 a^4+5 a^2+ a^{-2} +5
The A2 invariant q^{18}-q^{16}+q^{14}+3 q^{12}-5 q^{10}+3 q^8-3 q^6-2 q^4+4 q^2-4+7 q^{-2} -3 q^{-4} +2 q^{-6} +3 q^{-8} -4 q^{-10} +2 q^{-12} - q^{-14}
The G2 invariant Data:K11a274/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: (-1, 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
-4 16 8 -\frac{62}{3} -\frac{34}{3} -64 -\frac{320}{3} -\frac{224}{3} 16 -\frac{32}{3} 128 \frac{248}{3} \frac{136}{3} \frac{11729}{30} -\frac{2258}{15} \frac{18658}{45} -\frac{689}{18} \frac{2609}{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=0 is the signature of K11a274. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012345χ
11           1-1
9          3 3
7         71 -6
5        113  8
3       127   -5
1      1511    4
-1     1313     0
-3    1014      -4
-5   713       6
-7  310        -7
-9 17         6
-11 3          -3
-131           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{15}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\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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K11a273

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K11a275