K11n174

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K11n173.gif

K11n173

K11n175.gif

K11n175

Contents

K11n174.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n174 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X3,11,4,10 X16,6,17,5 X14,7,15,8 X20,10,21,9 X11,5,12,4 X18,14,19,13 X2,16,3,15 X22,17,1,18 X8,20,9,19 X12,22,13,21
Gauss code 1, -8, -2, 6, 3, -1, 4, -10, 5, 2, -6, -11, 7, -4, 8, -3, 9, -7, 10, -5, 11, -9
Dowker-Thistlethwaite code 6 -10 16 14 20 -4 18 2 22 8 12
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gif
A Morse Link Presentation K11n174 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n174/ThurstonBennequinNumber
Hyperbolic Volume 16.7284
A-Polynomial See Data:K11n174/A-polynomial

[edit Notes for K11n174's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n174's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+11 t^2-22 t+27-22 t^{-1} +11 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 97, 4 }
Jones polynomial -q^{11}+4 q^{10}-9 q^9+13 q^8-16 q^7+17 q^6-15 q^5+12 q^4-7 q^3+3 q^2
HOMFLY-PT polynomial (db, data sources) -2 z^6 a^{-6} +3 z^4 a^{-4} -7 z^4 a^{-6} +3 z^4 a^{-8} +7 z^2 a^{-4} -8 z^2 a^{-6} +6 z^2 a^{-8} -z^2 a^{-10} +3 a^{-4} -3 a^{-6} +2 a^{-8} - a^{-10}
Kauffman polynomial (db, data sources) 3 z^9 a^{-7} +3 z^9 a^{-9} +6 z^8 a^{-6} +14 z^8 a^{-8} +8 z^8 a^{-10} +3 z^7 a^{-5} +4 z^7 a^{-7} +9 z^7 a^{-9} +8 z^7 a^{-11} -12 z^6 a^{-6} -28 z^6 a^{-8} -12 z^6 a^{-10} +4 z^6 a^{-12} -12 z^5 a^{-7} -27 z^5 a^{-9} -14 z^5 a^{-11} +z^5 a^{-13} +6 z^4 a^{-4} +18 z^4 a^{-6} +22 z^4 a^{-8} +5 z^4 a^{-10} -5 z^4 a^{-12} -2 z^3 a^{-5} +7 z^3 a^{-7} +18 z^3 a^{-9} +8 z^3 a^{-11} -z^3 a^{-13} -9 z^2 a^{-4} -14 z^2 a^{-6} -9 z^2 a^{-8} -3 z^2 a^{-10} +z^2 a^{-12} -z a^{-5} -z a^{-7} -3 z a^{-9} -3 z a^{-11} +3 a^{-4} +3 a^{-6} +2 a^{-8} + a^{-10}
The A2 invariant 3 q^{-6} -2 q^{-8} +3 q^{-10} -2 q^{-14} +4 q^{-16} -3 q^{-18} +3 q^{-20} -2 q^{-22} - q^{-24} +2 q^{-26} -3 q^{-28} +2 q^{-30} - q^{-34}
The G2 invariant Data:K11n174/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a64,}

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

Vassiliev invariants

V2 and V3: (4, 9)
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
16 72 128 \frac{1208}{3} \frac{208}{3} 1152 2416 416 392 \frac{2048}{3} 2592 \frac{19328}{3} \frac{3328}{3} \frac{219182}{15} -\frac{1216}{5} \frac{296528}{45} \frac{1666}{9} \frac{13262}{15}

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=4 is the signature of K11n174. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
0123456789χ
23         1-1
21        3 3
19       61 -5
17      73  4
15     96   -3
13    87    1
11   79     2
9  58      -3
7 27       5
515        -4
33         3
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=0 {\mathbb Z}^{3} {\mathbb Z}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=9 {\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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K11n173.gif

K11n173

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K11n175