K11a22

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K11a21.gif

K11a21

K11a23.gif

K11a23

Contents

K11a22.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a22 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X12,5,13,6 X2837 X18,10,19,9 X14,11,15,12 X6,13,7,14 X20,16,21,15 X22,18,1,17 X10,20,11,19 X16,22,17,21
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -3, 7, -6, 8, -11, 9, -5, 10, -8, 11, -9
Dowker-Thistlethwaite code 4 8 12 2 18 14 6 20 22 10 16
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation K11a22 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a22/ThurstonBennequinNumber
Hyperbolic Volume 14.6533
A-Polynomial See Data:K11a22/A-polynomial

[edit Notes for K11a22'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 K11a22's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+13 t^2-20 t+23-20 t^{-1} +13 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6+3 z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 101, 4 }
Jones polynomial q^{10}-3 q^9+6 q^8-11 q^7+14 q^6-16 q^5+16 q^4-13 q^3+11 q^2-6 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} +15 z^4 a^{-4} -9 z^4 a^{-6} +z^4 a^{-8} -5 z^2 a^{-2} +19 z^2 a^{-4} -14 z^2 a^{-6} +3 z^2 a^{-8} -2 a^{-2} +9 a^{-4} -8 a^{-6} +2 a^{-8}
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} +8 z^8 a^{-4} +12 z^8 a^{-6} +7 z^8 a^{-8} +z^7 a^{-1} -6 z^7 a^{-3} -11 z^7 a^{-5} +3 z^7 a^{-7} +7 z^7 a^{-9} -12 z^6 a^{-2} -38 z^6 a^{-4} -41 z^6 a^{-6} -10 z^6 a^{-8} +5 z^6 a^{-10} -4 z^5 a^{-1} -7 z^5 a^{-3} -17 z^5 a^{-5} -26 z^5 a^{-7} -9 z^5 a^{-9} +3 z^5 a^{-11} +16 z^4 a^{-2} +48 z^4 a^{-4} +43 z^4 a^{-6} +6 z^4 a^{-8} -4 z^4 a^{-10} +z^4 a^{-12} +5 z^3 a^{-1} +17 z^3 a^{-3} +32 z^3 a^{-5} +29 z^3 a^{-7} +6 z^3 a^{-9} -3 z^3 a^{-11} -9 z^2 a^{-2} -29 z^2 a^{-4} -25 z^2 a^{-6} -3 z^2 a^{-8} +z^2 a^{-10} -z^2 a^{-12} -2 z a^{-1} -7 z a^{-3} -14 z a^{-5} -11 z a^{-7} -z a^{-9} +z a^{-11} +2 a^{-2} +9 a^{-4} +8 a^{-6} +2 a^{-8}
The A2 invariant -q^2+1-2 q^{-2} + q^{-4} +2 q^{-6} +6 q^{-10} - q^{-12} +3 q^{-14} - q^{-16} -3 q^{-18} -4 q^{-22} + q^{-24} + q^{-30}
The G2 invariant q^{12}-2 q^{10}+6 q^8-11 q^6+14 q^4-16 q^2+5+17 q^{-2} -49 q^{-4} +82 q^{-6} -97 q^{-8} +72 q^{-10} -7 q^{-12} -91 q^{-14} +185 q^{-16} -233 q^{-18} +204 q^{-20} -95 q^{-22} -72 q^{-24} +227 q^{-26} -309 q^{-28} +285 q^{-30} -147 q^{-32} -33 q^{-34} +193 q^{-36} -259 q^{-38} +215 q^{-40} -72 q^{-42} -87 q^{-44} +211 q^{-46} -220 q^{-48} +128 q^{-50} +45 q^{-52} -215 q^{-54} +321 q^{-56} -304 q^{-58} +173 q^{-60} +35 q^{-62} -247 q^{-64} +384 q^{-66} -395 q^{-68} +272 q^{-70} -61 q^{-72} -166 q^{-74} +309 q^{-76} -336 q^{-78} +227 q^{-80} -51 q^{-82} -122 q^{-84} +211 q^{-86} -194 q^{-88} +76 q^{-90} +69 q^{-92} -183 q^{-94} +208 q^{-96} -141 q^{-98} +7 q^{-100} +127 q^{-102} -216 q^{-104} +231 q^{-106} -168 q^{-108} +63 q^{-110} +49 q^{-112} -134 q^{-114} +170 q^{-116} -158 q^{-118} +115 q^{-120} -46 q^{-122} -13 q^{-124} +59 q^{-126} -83 q^{-128} +81 q^{-130} -64 q^{-132} +40 q^{-134} -11 q^{-136} -11 q^{-138} +25 q^{-140} -30 q^{-142} +26 q^{-144} -18 q^{-146} +10 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: {}

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

Vassiliev invariants

V2 and V3: (3, 3)
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
12 24 72 78 10 288 304 64 56 288 288 936 120 \frac{11871}{10} -\frac{1186}{15} \frac{10502}{15} \frac{65}{6} \frac{991}{10}

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 K11a22. 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        72  -5
13       74   3
11      97    -2
9     77     0
7    69      3
5   57       -2
3  27        5
1 14         -3
-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}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
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}^{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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