K11a217

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K11a216.gif

K11a216

K11a218.gif

K11a218

Contents

K11a217.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a217 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X18,5,19,6 X14,8,15,7 X20,9,21,10 X16,12,17,11 X2,13,3,14 X8,16,9,15 X22,17,1,18 X6,19,7,20 X10,21,11,22
Gauss code 1, -7, 2, -1, 3, -10, 4, -8, 5, -11, 6, -2, 7, -4, 8, -6, 9, -3, 10, -5, 11, -9
Dowker-Thistlethwaite code 4 12 18 14 20 16 2 8 22 6 10
A Braid Representative
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BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation K11a217 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a217/ThurstonBennequinNumber
Hyperbolic Volume 17.2462
A-Polynomial See Data:K11a217/A-polynomial

[edit Notes for K11a217's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant 2

[edit Notes for K11a217's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+6 t^3-16 t^2+29 t-35+29 t^{-1} -16 t^{-2} +6 t^{-3} - t^{-4}
Conway polynomial -z^8-2 z^6+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 139, -2 }
Jones polynomial q^3-4 q^2+9 q-15+20 q^{-1} -22 q^{-2} +23 q^{-3} -19 q^{-4} +14 q^{-5} -8 q^{-6} +3 q^{-7} - q^{-8}
HOMFLY-PT polynomial (db, data sources) -a^2 z^8+2 a^4 z^6-5 a^2 z^6+z^6-a^6 z^4+8 a^4 z^4-10 a^2 z^4+3 z^4-3 a^6 z^2+12 a^4 z^2-9 a^2 z^2+3 z^2-3 a^6+6 a^4-3 a^2+1
Kauffman polynomial (db, data sources) 2 a^4 z^{10}+2 a^2 z^{10}+6 a^5 z^9+12 a^3 z^9+6 a z^9+8 a^6 z^8+14 a^4 z^8+13 a^2 z^8+7 z^8+6 a^7 z^7-a^5 z^7-16 a^3 z^7-5 a z^7+4 z^7 a^{-1} +3 a^8 z^6-13 a^6 z^6-38 a^4 z^6-38 a^2 z^6+z^6 a^{-2} -15 z^6+a^9 z^5-9 a^7 z^5-13 a^5 z^5-5 a^3 z^5-11 a z^5-9 z^5 a^{-1} -4 a^8 z^4+14 a^6 z^4+38 a^4 z^4+31 a^2 z^4-2 z^4 a^{-2} +9 z^4-2 a^9 z^3+7 a^7 z^3+18 a^5 z^3+13 a^3 z^3+10 a z^3+6 z^3 a^{-1} +a^8 z^2-9 a^6 z^2-20 a^4 z^2-14 a^2 z^2+z^2 a^{-2} -3 z^2+a^9 z-3 a^7 z-7 a^5 z-5 a^3 z-3 a z-z a^{-1} +3 a^6+6 a^4+3 a^2+1
The A2 invariant -q^{24}-q^{20}-3 q^{18}+4 q^{16}-2 q^{14}+3 q^{12}+3 q^{10}-2 q^8+5 q^6-5 q^4+3 q^2-1-2 q^{-2} +3 q^{-4} -2 q^{-6} + q^{-8}
The G2 invariant q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+10 q^{120}-10 q^{118}+4 q^{116}+10 q^{114}-29 q^{112}+52 q^{110}-73 q^{108}+76 q^{106}-58 q^{104}+5 q^{102}+83 q^{100}-182 q^{98}+271 q^{96}-306 q^{94}+242 q^{92}-75 q^{90}-188 q^{88}+468 q^{86}-660 q^{84}+663 q^{82}-438 q^{80}+11 q^{78}+468 q^{76}-819 q^{74}+889 q^{72}-622 q^{70}+115 q^{68}+415 q^{66}-748 q^{64}+719 q^{62}-337 q^{60}-214 q^{58}+689 q^{56}-838 q^{54}+581 q^{52}-653 q^{48}+1119 q^{46}-1184 q^{44}+816 q^{42}-131 q^{40}-614 q^{38}+1165 q^{36}-1312 q^{34}+1024 q^{32}-406 q^{30}-307 q^{28}+840 q^{26}-1013 q^{24}+772 q^{22}-239 q^{20}-347 q^{18}+723 q^{16}-730 q^{14}+362 q^{12}+203 q^{10}-707 q^8+918 q^6-745 q^4+257 q^2+326-774 q^{-2} +931 q^{-4} -752 q^{-6} +356 q^{-8} +92 q^{-10} -432 q^{-12} +563 q^{-14} -494 q^{-16} +304 q^{-18} -82 q^{-20} -88 q^{-22} +171 q^{-24} -178 q^{-26} +131 q^{-28} -67 q^{-30} +18 q^{-32} +14 q^{-34} -25 q^{-36} +22 q^{-38} -16 q^{-40} +8 q^{-42} -3 q^{-44} + q^{-46}

"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, -6)
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 -48 72 206 34 -576 -1056 -128 -208 288 1152 2472 408 \frac{54591}{10} -\frac{1022}{5} \frac{36782}{15} \frac{737}{6} \frac{3231}{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=-2 is the signature of K11a217. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-7-6-5-4-3-2-101234χ
7           11
5          3 -3
3         61 5
1        93  -6
-1       116   5
-3      1210    -2
-5     1110     1
-7    812      4
-9   611       -5
-11  28        6
-13 16         -5
-15 2          2
-171           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\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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K11a216.gif

K11a216

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K11a218