K11a40

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K11a39.gif

K11a39

K11a41.gif

K11a41

Contents

K11a40.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a40 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X14,5,15,6 X2837 X18,10,19,9 X20,12,21,11 X16,13,17,14 X6,15,7,16 X22,18,1,17 X10,20,11,19 X12,22,13,21
Gauss code 1, -4, 2, -1, 3, -8, 4, -2, 5, -10, 6, -11, 7, -3, 8, -7, 9, -5, 10, -6, 11, -9
Dowker-Thistlethwaite code 4 8 14 2 18 20 16 6 22 10 12
A Braid Representative
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A Morse Link Presentation K11a40 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a40/ThurstonBennequinNumber
Hyperbolic Volume 12.8867
A-Polynomial See Data:K11a40/A-polynomial

[edit Notes for K11a40'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 K11a40's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+12 t^2-17 t+19-17 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6+2 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 89, 4 }
Jones polynomial q^{10}-3 q^9+6 q^8-10 q^7+12 q^6-14 q^5+14 q^4-11 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} -9 z^4 a^{-6} +z^4 a^{-8} -4 z^2 a^{-2} +16 z^2 a^{-4} -13 z^2 a^{-6} +3 z^2 a^{-8} - a^{-2} +7 a^{-4} -7 a^{-6} +2 a^{-8}
Kauffman polynomial (db, data sources) z^{10} a^{-4} +z^{10} a^{-6} +3 z^9 a^{-3} +6 z^9 a^{-5} +3 z^9 a^{-7} +3 z^8 a^{-2} +5 z^8 a^{-4} +7 z^8 a^{-6} +5 z^8 a^{-8} +z^7 a^{-1} -9 z^7 a^{-3} -14 z^7 a^{-5} +2 z^7 a^{-7} +6 z^7 a^{-9} -13 z^6 a^{-2} -30 z^6 a^{-4} -26 z^6 a^{-6} -4 z^6 a^{-8} +5 z^6 a^{-10} -4 z^5 a^{-1} +2 z^5 a^{-3} -16 z^5 a^{-7} -7 z^5 a^{-9} +3 z^5 a^{-11} +17 z^4 a^{-2} +42 z^4 a^{-4} +30 z^4 a^{-6} -z^4 a^{-8} -5 z^4 a^{-10} +z^4 a^{-12} +4 z^3 a^{-1} +8 z^3 a^{-3} +14 z^3 a^{-5} +17 z^3 a^{-7} +4 z^3 a^{-9} -3 z^3 a^{-11} -8 z^2 a^{-2} -25 z^2 a^{-4} -21 z^2 a^{-6} -z^2 a^{-8} +2 z^2 a^{-10} -z^2 a^{-12} -z a^{-1} -3 z a^{-3} -8 z a^{-5} -8 z a^{-7} -z a^{-9} +z a^{-11} + a^{-2} +7 a^{-4} +7 a^{-6} +2 a^{-8}
The A2 invariant -q^2+1- q^{-2} + q^{-4} +2 q^{-6} +5 q^{-10} - q^{-12} +2 q^{-14} - q^{-16} -3 q^{-18} -3 q^{-22} + q^{-24} + 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} -36 q^{-4} +57 q^{-6} -65 q^{-8} +46 q^{-10} -4 q^{-12} -58 q^{-14} +118 q^{-16} -146 q^{-18} +129 q^{-20} -62 q^{-22} -37 q^{-24} +130 q^{-26} -180 q^{-28} +171 q^{-30} -98 q^{-32} + q^{-34} +95 q^{-36} -139 q^{-38} +127 q^{-40} -58 q^{-42} -22 q^{-44} +94 q^{-46} -112 q^{-48} +76 q^{-50} +5 q^{-52} -92 q^{-54} +161 q^{-56} -166 q^{-58} +111 q^{-60} -6 q^{-62} -115 q^{-64} +201 q^{-66} -231 q^{-68} +181 q^{-70} -73 q^{-72} -59 q^{-74} +157 q^{-76} -198 q^{-78} +162 q^{-80} -78 q^{-82} -24 q^{-84} +87 q^{-86} -106 q^{-88} +69 q^{-90} -6 q^{-92} -55 q^{-94} +86 q^{-96} -73 q^{-98} +23 q^{-100} +37 q^{-102} -92 q^{-104} +116 q^{-106} -103 q^{-108} +64 q^{-110} -6 q^{-112} -52 q^{-114} +97 q^{-116} -115 q^{-118} +106 q^{-120} -67 q^{-122} +20 q^{-124} +27 q^{-126} -63 q^{-128} +77 q^{-130} -70 q^{-132} +51 q^{-134} -20 q^{-136} -6 q^{-138} +23 q^{-140} -31 q^{-142} +28 q^{-144} -20 q^{-146} +11 q^{-148} -2 q^{-150} -4 q^{-152} +5 q^{-154} -6 q^{-156} +4 q^{-158} -2 q^{-160} + q^{-162}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a330,}

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

Vassiliev invariants

V2 and V3: (2, 1)
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
8 8 32 -\frac{20}{3} -\frac{4}{3} 64 -\frac{208}{3} \frac{32}{3} 8 \frac{256}{3} 32 -\frac{160}{3} -\frac{32}{3} -\frac{5129}{15} -\frac{2564}{15} \frac{11284}{45} -\frac{103}{9} \frac{631}{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 K11a40. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-1012345678χ
21           11
19          2 -2
17         41 3
15        62  -4
13       64   2
11      86    -2
9     66     0
7    58      3
5   46       -2
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}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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

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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K11a39.gif

K11a39

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K11a41