K11a215

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K11a214.gif

K11a214

K11a216.gif

K11a216

Contents

K11a215.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a215 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X18,6,19,5 X14,7,15,8 X16,10,17,9 X2,12,3,11 X22,13,1,14 X20,16,21,15 X10,18,11,17 X6,20,7,19 X8,21,9,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -11, 5, -9, 6, -2, 7, -4, 8, -5, 9, -3, 10, -8, 11, -7
Dowker-Thistlethwaite code 4 12 18 14 16 2 22 20 10 6 8
A Braid Representative
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A Morse Link Presentation K11a215 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:K11a215/ThurstonBennequinNumber
Hyperbolic Volume 16.7322
A-Polynomial See Data:K11a215/A-polynomial

[edit Notes for K11a215'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 K11a215's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-6 t^3+17 t^2-27 t+31-27 t^{-1} +17 t^{-2} -6 t^{-3} + t^{-4}
Conway polynomial z^8+2 z^6+z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 133, 4 }
Jones polynomial q^{10}-4 q^9+8 q^8-14 q^7+19 q^6-21 q^5+21 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} -7 z^2 a^{-6} +2 z^2 a^{-8} +3 a^{-4} -2 a^{-6}
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} +19 z^8 a^{-6} +13 z^8 a^{-8} +z^7 a^{-1} -12 z^7 a^{-3} -27 z^7 a^{-5} -2 z^7 a^{-7} +12 z^7 a^{-9} -14 z^6 a^{-2} -49 z^6 a^{-4} -64 z^6 a^{-6} -21 z^6 a^{-8} +8 z^6 a^{-10} -3 z^5 a^{-1} +z^5 a^{-3} -3 z^5 a^{-5} -26 z^5 a^{-7} -15 z^5 a^{-9} +4 z^5 a^{-11} +16 z^4 a^{-2} +54 z^4 a^{-4} +56 z^4 a^{-6} +11 z^4 a^{-8} -6 z^4 a^{-10} +z^4 a^{-12} +3 z^3 a^{-1} +10 z^3 a^{-3} +22 z^3 a^{-5} +23 z^3 a^{-7} +6 z^3 a^{-9} -2 z^3 a^{-11} -6 z^2 a^{-2} -21 z^2 a^{-4} -18 z^2 a^{-6} -2 z^2 a^{-8} +z^2 a^{-10} -z a^{-1} -4 z a^{-3} -8 z a^{-5} -5 z a^{-7} +3 a^{-4} +2 a^{-6}
The A2 invariant -q^2+2-2 q^{-2} +2 q^{-4} +2 q^{-6} -2 q^{-8} +5 q^{-10} -4 q^{-12} +3 q^{-14} - q^{-18} +3 q^{-20} -4 q^{-22} + q^{-24} - q^{-26} - 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} +165 q^{-6} -205 q^{-8} +166 q^{-10} -35 q^{-12} -176 q^{-14} +395 q^{-16} -518 q^{-18} +476 q^{-20} -232 q^{-22} -146 q^{-24} +522 q^{-26} -735 q^{-28} +685 q^{-30} -371 q^{-32} -82 q^{-34} +483 q^{-36} -670 q^{-38} +562 q^{-40} -208 q^{-42} -215 q^{-44} +535 q^{-46} -584 q^{-48} +345 q^{-50} +76 q^{-52} -513 q^{-54} +774 q^{-56} -745 q^{-58} +424 q^{-60} +89 q^{-62} -602 q^{-64} +937 q^{-66} -952 q^{-68} +640 q^{-70} -115 q^{-72} -433 q^{-74} +789 q^{-76} -837 q^{-78} +567 q^{-80} -103 q^{-82} -338 q^{-84} +580 q^{-86} -525 q^{-88} +214 q^{-90} +182 q^{-92} -488 q^{-94} +556 q^{-96} -372 q^{-98} +23 q^{-100} +335 q^{-102} -564 q^{-104} +587 q^{-106} -407 q^{-108} +116 q^{-110} +170 q^{-112} -372 q^{-114} +426 q^{-116} -361 q^{-118} +225 q^{-120} -58 q^{-122} -76 q^{-124} +164 q^{-126} -192 q^{-128} +168 q^{-130} -118 q^{-132} +59 q^{-134} -5 q^{-136} -35 q^{-138} +55 q^{-140} -60 q^{-142} +48 q^{-144} -29 q^{-146} +14 q^{-148} + q^{-150} -8 q^{-152} +10 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: (3, 5)
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 40 72 174 26 480 \frac{2608}{3} \frac{544}{3} 104 288 800 2088 312 \frac{42751}{10} \frac{2734}{15} \frac{8434}{5} \frac{11}{2} \frac{2431}{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 K11a215. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-1012345678χ
21           11
19          3 -3
17         51 4
15        93  -6
13       105   5
11      119    -2
9     1010     0
7    811      3
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}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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.

See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

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K11a214.gif

K11a214

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K11a216