K11a177

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K11a176.gif

K11a176

K11a178.gif

K11a178

Contents

K11a177.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a177 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X14,5,15,6 X16,8,17,7 X18,10,19,9 X2,12,3,11 X22,13,1,14 X20,16,21,15 X8,18,9,17 X10,20,11,19 X6,21,7,22
Gauss code 1, -6, 2, -1, 3, -11, 4, -9, 5, -10, 6, -2, 7, -3, 8, -4, 9, -5, 10, -8, 11, -7
Dowker-Thistlethwaite code 4 12 14 16 18 2 22 20 8 10 6
A Braid Representative
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A Morse Link Presentation K11a177 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 4
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a177/ThurstonBennequinNumber
Hyperbolic Volume 13.045
A-Polynomial See Data:K11a177/A-polynomial

[edit Notes for K11a177's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a177's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+13 t^2-19 t+21-19 t^{-1} +13 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6+3 z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 97, 4 }
Jones polynomial q^{10}-4 q^9+7 q^8-11 q^7+14 q^6-15 q^5+15 q^4-12 q^3+9 q^2-5 q+3- q^{-1}
HOMFLY-PT polynomial (db, data sources) z^8 a^{-4} -z^6 a^{-2} +6 z^6 a^{-4} -2 z^6 a^{-6} -4 z^4 a^{-2} +14 z^4 a^{-4} -8 z^4 a^{-6} +z^4 a^{-8} -4 z^2 a^{-2} +15 z^2 a^{-4} -9 z^2 a^{-6} +2 z^2 a^{-8} - a^{-2} +5 a^{-4} -3 a^{-6}
Kauffman polynomial (db, data sources) z^{10} a^{-4} +z^{10} a^{-6} +3 z^9 a^{-3} +7 z^9 a^{-5} +4 z^9 a^{-7} +3 z^8 a^{-2} +6 z^8 a^{-4} +10 z^8 a^{-6} +7 z^8 a^{-8} +z^7 a^{-1} -9 z^7 a^{-3} -18 z^7 a^{-5} +8 z^7 a^{-9} -13 z^6 a^{-2} -34 z^6 a^{-4} -36 z^6 a^{-6} -8 z^6 a^{-8} +7 z^6 a^{-10} -4 z^5 a^{-1} +3 z^5 a^{-3} +7 z^5 a^{-5} -12 z^5 a^{-7} -8 z^5 a^{-9} +4 z^5 a^{-11} +17 z^4 a^{-2} +47 z^4 a^{-4} +39 z^4 a^{-6} +z^4 a^{-8} -7 z^4 a^{-10} +z^4 a^{-12} +4 z^3 a^{-1} +6 z^3 a^{-3} +8 z^3 a^{-5} +8 z^3 a^{-7} -z^3 a^{-9} -3 z^3 a^{-11} -8 z^2 a^{-2} -25 z^2 a^{-4} -18 z^2 a^{-6} +z^2 a^{-10} -z a^{-1} -3 z a^{-3} -5 z a^{-5} -z a^{-7} +2 z a^{-9} + a^{-2} +5 a^{-4} +3 a^{-6}
The A2 invariant -q^2+1- q^{-2} + q^{-4} +2 q^{-6} - q^{-8} +4 q^{-10} -2 q^{-12} +2 q^{-14} + q^{-16} - q^{-18} +2 q^{-20} -3 q^{-22} - q^{-26} - q^{-28} + q^{-30}
The G2 invariant q^{12}-2 q^{10}+5 q^8-9 q^6+10 q^4-11 q^2+3+14 q^{-2} -35 q^{-4} +56 q^{-6} -66 q^{-8} +50 q^{-10} -12 q^{-12} -51 q^{-14} +116 q^{-16} -156 q^{-18} +154 q^{-20} -93 q^{-22} -8 q^{-24} +123 q^{-26} -205 q^{-28} +225 q^{-30} -168 q^{-32} +57 q^{-34} +71 q^{-36} -167 q^{-38} +196 q^{-40} -142 q^{-42} +46 q^{-44} +69 q^{-46} -136 q^{-48} +131 q^{-50} -62 q^{-52} -50 q^{-54} +153 q^{-56} -199 q^{-58} +170 q^{-60} -60 q^{-62} -87 q^{-64} +224 q^{-66} -291 q^{-68} +264 q^{-70} -149 q^{-72} -16 q^{-74} +171 q^{-76} -265 q^{-78} +267 q^{-80} -178 q^{-82} +44 q^{-84} +84 q^{-86} -158 q^{-88} +150 q^{-90} -84 q^{-92} -12 q^{-94} +86 q^{-96} -112 q^{-98} +76 q^{-100} - q^{-102} -88 q^{-104} +148 q^{-106} -157 q^{-108} +113 q^{-110} -33 q^{-112} -62 q^{-114} +131 q^{-116} -166 q^{-118} +158 q^{-120} -107 q^{-122} +37 q^{-124} +37 q^{-126} -91 q^{-128} +114 q^{-130} -107 q^{-132} +78 q^{-134} -35 q^{-136} -6 q^{-138} +34 q^{-140} -50 q^{-142} +47 q^{-144} -32 q^{-146} +18 q^{-148} -2 q^{-150} -6 q^{-152} +9 q^{-154} -10 q^{-156} +6 q^{-158} -3 q^{-160} + q^{-162}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (4, 7)
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 56 128 \frac{824}{3} \frac{112}{3} 896 \frac{4496}{3} \frac{896}{3} 152 \frac{2048}{3} 1568 \frac{13184}{3} \frac{1792}{3} \frac{124142}{15} \frac{8032}{15} \frac{129968}{45} \frac{34}{9} \frac{5582}{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 K11a177. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-1012345678χ
21           11
19          3 -3
17         41 3
15        73  -4
13       74   3
11      87    -1
9     77     0
7    58      3
5   47       -3
3  26        4
1 13         -2
-1 2          2
-31           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\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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K11a176

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K11a178