K11a127

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K11a126.gif

K11a126

K11a128.gif

K11a128

Contents

K11a127.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a127 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,5,15,6 X20,8,21,7 X2,10,3,9 X22,11,1,12 X18,14,19,13 X8,15,9,16 X12,18,13,17 X6,20,7,19 X16,22,17,21
Gauss code 1, -5, 2, -1, 3, -10, 4, -8, 5, -2, 6, -9, 7, -3, 8, -11, 9, -7, 10, -4, 11, -6
Dowker-Thistlethwaite code 4 10 14 20 2 22 18 8 12 6 16
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11a127 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:K11a127/ThurstonBennequinNumber
Hyperbolic Volume 16.6559
A-Polynomial See Data:K11a127/A-polynomial

[edit Notes for K11a127'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 K11a127's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-6 t^3+17 t^2-28 t+33-28 t^{-1} +17 t^{-2} -6 t^{-3} + t^{-4}
Conway polynomial z^8+2 z^6+z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 137, 4 }
Jones polynomial q^{10}-4 q^9+9 q^8-15 q^7+19 q^6-22 q^5+22 q^4-18 q^3+14 q^2-8 q+4- q^{-1}
HOMFLY-PT polynomial (db, data sources) z^8 a^{-4} -z^6 a^{-2} +5 z^6 a^{-4} -2 z^6 a^{-6} -3 z^4 a^{-2} +10 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} -2 z^2 a^{-2} +10 z^2 a^{-4} -8 z^2 a^{-6} +2 z^2 a^{-8} +4 a^{-4} -4 a^{-6} + a^{-8}
Kauffman polynomial (db, data sources) 2 z^{10} a^{-4} +2 z^{10} a^{-6} +5 z^9 a^{-3} +13 z^9 a^{-5} +8 z^9 a^{-7} +4 z^8 a^{-2} +10 z^8 a^{-4} +20 z^8 a^{-6} +14 z^8 a^{-8} +z^7 a^{-1} -12 z^7 a^{-3} -25 z^7 a^{-5} +2 z^7 a^{-7} +14 z^7 a^{-9} -14 z^6 a^{-2} -48 z^6 a^{-4} -64 z^6 a^{-6} -21 z^6 a^{-8} +9 z^6 a^{-10} -3 z^5 a^{-1} +z^5 a^{-3} -8 z^5 a^{-5} -36 z^5 a^{-7} -20 z^5 a^{-9} +4 z^5 a^{-11} +16 z^4 a^{-2} +53 z^4 a^{-4} +54 z^4 a^{-6} +9 z^4 a^{-8} -7 z^4 a^{-10} +z^4 a^{-12} +3 z^3 a^{-1} +11 z^3 a^{-3} +27 z^3 a^{-5} +32 z^3 a^{-7} +12 z^3 a^{-9} -z^3 a^{-11} -6 z^2 a^{-2} -21 z^2 a^{-4} -20 z^2 a^{-6} -3 z^2 a^{-8} +2 z^2 a^{-10} -z a^{-1} -4 z a^{-3} -10 z a^{-5} -10 z a^{-7} -3 z a^{-9} +4 a^{-4} +4 a^{-6} + a^{-8}
The A2 invariant -q^2+2-2 q^{-2} +2 q^{-4} +2 q^{-6} -2 q^{-8} +6 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} -3 q^{-18} +2 q^{-20} -4 q^{-22} +2 q^{-24} - q^{-28} + q^{-30}
The G2 invariant q^{12}-3 q^{10}+9 q^8-19 q^6+28 q^4-33 q^2+17+26 q^{-2} -91 q^{-4} +164 q^{-6} -204 q^{-8} +168 q^{-10} -41 q^{-12} -167 q^{-14} +393 q^{-16} -528 q^{-18} +497 q^{-20} -263 q^{-22} -119 q^{-24} +510 q^{-26} -760 q^{-28} +752 q^{-30} -457 q^{-32} -4 q^{-34} +460 q^{-36} -713 q^{-38} +662 q^{-40} -333 q^{-42} -116 q^{-44} +492 q^{-46} -624 q^{-48} +450 q^{-50} -37 q^{-52} -434 q^{-54} +775 q^{-56} -811 q^{-58} +524 q^{-60} + q^{-62} -577 q^{-64} +976 q^{-66} -1060 q^{-68} +779 q^{-70} -227 q^{-72} -388 q^{-74} +840 q^{-76} -967 q^{-78} +728 q^{-80} -250 q^{-82} -270 q^{-84} +594 q^{-86} -624 q^{-88} +356 q^{-90} +62 q^{-92} -428 q^{-94} +588 q^{-96} -465 q^{-98} +124 q^{-100} +274 q^{-102} -578 q^{-104} +662 q^{-106} -511 q^{-108} +211 q^{-110} +135 q^{-112} -399 q^{-114} +517 q^{-116} -473 q^{-118} +312 q^{-120} -101 q^{-122} -88 q^{-124} +207 q^{-126} -253 q^{-128} +225 q^{-130} -153 q^{-132} +72 q^{-134} +6 q^{-136} -54 q^{-138} +72 q^{-140} -73 q^{-142} +54 q^{-144} -31 q^{-146} +12 q^{-148} +4 q^{-150} -10 q^{-152} +11 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: (2, 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
8 16 32 \frac{124}{3} \frac{20}{3} 128 \frac{448}{3} \frac{160}{3} 16 \frac{256}{3} 128 \frac{992}{3} \frac{160}{3} \frac{7471}{15} \frac{356}{15} \frac{13324}{45} -\frac{127}{9} \frac{991}{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 K11a127. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-1012345678χ
21           11
19          3 -3
17         61 5
15        93  -6
13       106   4
11      129    -3
9     1010     0
7    812      4
5   610       -4
3  39        6
1 15         -4
-1 3          3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
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

Read me first: Modifying Knot Pages.

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K11a126

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K11a128