K11a88

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K11a87.gif

K11a87

K11a89.gif

K11a89

Contents

K11a88.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a88 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X16,7,17,8 X18,9,19,10 X2,11,3,12 X20,14,21,13 X22,16,1,15 X8,17,9,18 X6,19,7,20 X14,22,15,21
Gauss code 1, -6, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 10 12 16 18 2 20 22 8 6 14
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11a88 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a88/ThurstonBennequinNumber
Hyperbolic Volume 14.1444
A-Polynomial See Data:K11a88/A-polynomial

[edit Notes for K11a88's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant 0

[edit Notes for K11a88's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+12 t^2-20 t+25-20 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6+2 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 101, 0 }
Jones polynomial -q^5+3 q^4-6 q^3+11 q^2-14 q+16-16 q^{-1} +14 q^{-2} -10 q^{-3} +6 q^{-4} -3 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) z^8-2 a^2 z^6-z^6 a^{-2} +6 z^6+a^4 z^4-9 a^2 z^4-4 z^4 a^{-2} +14 z^4+3 a^4 z^2-13 a^2 z^2-5 z^2 a^{-2} +14 z^2+2 a^4-5 a^2- a^{-2} +5
Kauffman polynomial (db, data sources) a^2 z^{10}+z^{10}+3 a^3 z^9+7 a z^9+4 z^9 a^{-1} +4 a^4 z^8+7 a^2 z^8+6 z^8 a^{-2} +9 z^8+3 a^5 z^7-3 a^3 z^7-16 a z^7-5 z^7 a^{-1} +5 z^7 a^{-3} +a^6 z^6-10 a^4 z^6-25 a^2 z^6-14 z^6 a^{-2} +3 z^6 a^{-4} -31 z^6-9 a^5 z^5-4 a^3 z^5+18 a z^5+2 z^5 a^{-1} -10 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+7 a^4 z^4+33 a^2 z^4+16 z^4 a^{-2} -6 z^4 a^{-4} +45 z^4+7 a^5 z^3+2 a^3 z^3-10 a z^3+3 z^3 a^{-1} +6 z^3 a^{-3} -2 z^3 a^{-5} +2 a^6 z^2-4 a^4 z^2-23 a^2 z^2-7 z^2 a^{-2} +2 z^2 a^{-4} -26 z^2-a^5 z+a z-z a^{-1} -z a^{-3} +2 a^4+5 a^2+ a^{-2} +5
The A2 invariant q^{18}+q^{12}-3 q^{10}+2 q^8-q^6-q^4+2 q^2-3+4 q^{-2} - q^{-4} +2 q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12} - q^{-14}
The G2 invariant q^{94}-2 q^{92}+5 q^{90}-9 q^{88}+10 q^{86}-10 q^{84}+2 q^{82}+15 q^{80}-34 q^{78}+55 q^{76}-63 q^{74}+49 q^{72}-15 q^{70}-43 q^{68}+110 q^{66}-157 q^{64}+169 q^{62}-122 q^{60}+26 q^{58}+100 q^{56}-212 q^{54}+273 q^{52}-249 q^{50}+137 q^{48}+23 q^{46}-180 q^{44}+267 q^{42}-248 q^{40}+141 q^{38}+20 q^{36}-159 q^{34}+209 q^{32}-161 q^{30}+8 q^{28}+159 q^{26}-270 q^{24}+270 q^{22}-147 q^{20}-56 q^{18}+264 q^{16}-395 q^{14}+397 q^{12}-269 q^{10}+41 q^8+194 q^6-357 q^4+399 q^2-297+117 q^{-2} +85 q^{-4} -224 q^{-6} +254 q^{-8} -171 q^{-10} +19 q^{-12} +132 q^{-14} -205 q^{-16} +178 q^{-18} -50 q^{-20} -109 q^{-22} +236 q^{-24} -274 q^{-26} +216 q^{-28} -84 q^{-30} -82 q^{-32} +206 q^{-34} -259 q^{-36} +232 q^{-38} -136 q^{-40} +24 q^{-42} +70 q^{-44} -129 q^{-46} +139 q^{-48} -114 q^{-50} +67 q^{-52} -16 q^{-54} -22 q^{-56} +41 q^{-58} -45 q^{-60} +37 q^{-62} -23 q^{-64} +11 q^{-66} + q^{-68} -7 q^{-70} +7 q^{-72} -7 q^{-74} +4 q^{-76} -2 q^{-78} + q^{-80}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a84,}

Vassiliev invariants

V2 and V3: (-1, 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
-4 16 8 -\frac{62}{3} -\frac{34}{3} -64 -\frac{320}{3} -\frac{320}{3} 48 -\frac{32}{3} 128 \frac{248}{3} \frac{136}{3} \frac{11729}{30} -\frac{1778}{15} \frac{18658}{45} -\frac{1265}{18} \frac{2609}{30}

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=0 is the signature of K11a88. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012345χ
11           1-1
9          2 2
7         41 -3
5        72  5
3       74   -3
1      97    2
-1     88     0
-3    68      -2
-5   48       4
-7  26        -4
-9 14         3
-11 2          -2
-131           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\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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K11a87.gif

K11a87

K11a89.gif

K11a89