K11n177

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K11n176.gif

K11n176

K11n178.gif

K11n178

Contents

K11n177.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n177 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X3,11,4,10 X16,6,17,5 X18,7,19,8 X20,10,21,9 X11,3,12,2 X8,13,9,14 X4,16,5,15 X22,17,1,18 X12,20,13,19 X14,21,15,22
Gauss code 1, 6, -2, -8, 3, -1, 4, -7, 5, 2, -6, -10, 7, -11, 8, -3, 9, -4, 10, -5, 11, -9
Dowker-Thistlethwaite code 6 -10 16 18 20 -2 8 4 22 12 14
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation K11n177 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:K11n177/ThurstonBennequinNumber
Hyperbolic Volume 16.0917
A-Polynomial See Data:K11n177/A-polynomial

[edit Notes for K11n177's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n177's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+5 t^3-11 t^2+16 t-17+16 t^{-1} -11 t^{-2} +5 t^{-3} - t^{-4}
Conway polynomial -z^8-3 z^6-z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 83, 2 }
Jones polynomial -2 q^6+6 q^5-10 q^4+13 q^3-14 q^2+14 q-11+8 q^{-1} -4 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-2} -5 z^6 a^{-2} +z^6 a^{-4} +z^6-8 z^4 a^{-2} +4 z^4 a^{-4} +3 z^4-5 z^2 a^{-2} +5 z^2 a^{-4} -z^2 a^{-6} +2 z^2- a^{-2} +2 a^{-4} - a^{-6} +1
Kauffman polynomial (db, data sources) 3 z^9 a^{-1} +3 z^9 a^{-3} +13 z^8 a^{-2} +7 z^8 a^{-4} +6 z^8+4 a z^7+2 z^7 a^{-1} +3 z^7 a^{-3} +5 z^7 a^{-5} +a^2 z^6-34 z^6 a^{-2} -17 z^6 a^{-4} +z^6 a^{-6} -15 z^6-10 a z^5-21 z^5 a^{-1} -18 z^5 a^{-3} -7 z^5 a^{-5} -2 a^2 z^4+27 z^4 a^{-2} +23 z^4 a^{-4} +6 z^4 a^{-6} +8 z^4+6 a z^3+15 z^3 a^{-1} +16 z^3 a^{-3} +10 z^3 a^{-5} +3 z^3 a^{-7} +a^2 z^2-10 z^2 a^{-2} -12 z^2 a^{-4} -5 z^2 a^{-6} -2 z^2-a z-3 z a^{-1} -4 z a^{-3} -4 z a^{-5} -2 z a^{-7} + a^{-2} +2 a^{-4} + a^{-6} +1
The A2 invariant q^8-2 q^6+2 q^4-q^2+1+2 q^{-2} -3 q^{-4} +4 q^{-6} -3 q^{-8} +3 q^{-10} - q^{-14} +2 q^{-16} -2 q^{-18} + q^{-20} - q^{-22}
The G2 invariant Data:K11n177/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{158}{3} \frac{34}{3} 64 \frac{448}{3} \frac{64}{3} 16 \frac{32}{3} 128 \frac{632}{3} \frac{136}{3} \frac{16591}{30} \frac{726}{5} \frac{1502}{45} \frac{689}{18} -\frac{689}{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 K11n177. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-1012345χ
13         2-2
11        4 4
9       62 -4
7      74  3
5     76   -1
3    77    0
1   58     3
-1  36      -3
-3 15       4
-5 3        -3
-71         1
Integral Khovanov Homology

(db, data source)

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

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.

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K11n176.gif

K11n176

K11n178.gif

K11n178