K11a86

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K11a85.gif

K11a85

K11a87.gif

K11a87

Contents

K11a86.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a86 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X16,8,17,7 X18,10,19,9 X2,11,3,12 X20,13,21,14 X6,16,7,15 X8,18,9,17 X22,19,1,20 X14,21,15,22
Gauss code 1, -6, 2, -1, 3, -8, 4, -9, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, -7, 11, -10
Dowker-Thistlethwaite code 4 10 12 16 18 2 20 6 8 22 14
A Braid Representative
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A Morse Link Presentation K11a86 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:K11a86/ThurstonBennequinNumber
Hyperbolic Volume 13.5574
A-Polynomial See Data:K11a86/A-polynomial

[edit Notes for K11a86'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 K11a86's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+5 t^3-12 t^2+18 t-19+18 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 { 91, 2 }
Jones polynomial q^7-3 q^6+6 q^5-10 q^4+13 q^3-14 q^2+14 q-12+9 q^{-1} -5 q^{-2} +3 q^{-3} - q^{-4}
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-2} -6 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-14 z^4 a^{-2} +4 z^4 a^{-4} +9 z^4-3 a^2 z^2-15 z^2 a^{-2} +5 z^2 a^{-4} +12 z^2-a^2-5 a^{-2} +2 a^{-4} +5
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10}+3 a z^9+7 z^9 a^{-1} +4 z^9 a^{-3} +3 a^2 z^8+9 z^8 a^{-2} +6 z^8 a^{-4} +6 z^8+a^3 z^7-9 a z^7-20 z^7 a^{-1} -4 z^7 a^{-3} +6 z^7 a^{-5} -13 a^2 z^6-36 z^6 a^{-2} -9 z^6 a^{-4} +5 z^6 a^{-6} -35 z^6-4 a^3 z^5+3 a z^5+14 z^5 a^{-1} -2 z^5 a^{-3} -6 z^5 a^{-5} +3 z^5 a^{-7} +17 a^2 z^4+45 z^4 a^{-2} +7 z^4 a^{-4} -5 z^4 a^{-6} +z^4 a^{-8} +49 z^4+4 a^3 z^3+5 a z^3-2 z^3 a^{-1} +z^3 a^{-3} +z^3 a^{-5} -3 z^3 a^{-7} -8 a^2 z^2-26 z^2 a^{-2} -5 z^2 a^{-4} +2 z^2 a^{-6} -z^2 a^{-8} -26 z^2-a^3 z-2 a z-z a^{-1} +z a^{-5} +z a^{-7} +a^2+5 a^{-2} +2 a^{-4} +5
The A2 invariant -q^{12}+q^8+3 q^4-q^2+1+ q^{-2} -2 q^{-4} +3 q^{-6} -3 q^{-8} + q^{-10} - q^{-12} - q^{-14} +2 q^{-16} - q^{-18} + q^{-20}
The G2 invariant q^{60}-2 q^{58}+5 q^{56}-9 q^{54}+10 q^{52}-11 q^{50}+3 q^{48}+14 q^{46}-35 q^{44}+57 q^{42}-66 q^{40}+48 q^{38}-8 q^{36}-54 q^{34}+115 q^{32}-153 q^{30}+144 q^{28}-77 q^{26}-25 q^{24}+131 q^{22}-196 q^{20}+197 q^{18}-123 q^{16}+13 q^{14}+97 q^{12}-163 q^{10}+159 q^8-78 q^6-23 q^4+116 q^2-144+96 q^{-2} -2 q^{-4} -111 q^{-6} +188 q^{-8} -199 q^{-10} +135 q^{-12} -10 q^{-14} -126 q^{-16} +233 q^{-18} -263 q^{-20} +201 q^{-22} -81 q^{-24} -64 q^{-26} +176 q^{-28} -222 q^{-30} +186 q^{-32} -84 q^{-34} -29 q^{-36} +115 q^{-38} -137 q^{-40} +82 q^{-42} -82 q^{-46} +119 q^{-48} -98 q^{-50} +32 q^{-52} +51 q^{-54} -118 q^{-56} +147 q^{-58} -124 q^{-60} +63 q^{-62} +7 q^{-64} -72 q^{-66} +110 q^{-68} -116 q^{-70} +101 q^{-72} -58 q^{-74} +14 q^{-76} +30 q^{-78} -62 q^{-80} +71 q^{-82} -66 q^{-84} +48 q^{-86} -20 q^{-88} -4 q^{-90} +22 q^{-92} -30 q^{-94} +28 q^{-96} -20 q^{-98} +11 q^{-100} -2 q^{-102} -4 q^{-104} +5 q^{-106} -6 q^{-108} +4 q^{-110} -2 q^{-112} + q^{-114}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a205,}

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{34}{3} \frac{62}{3} 64 \frac{608}{3} \frac{224}{3} 80 -\frac{32}{3} 128 -\frac{136}{3} -\frac{248}{3} \frac{13169}{30} \frac{714}{5} \frac{898}{45} \frac{1999}{18} -\frac{2191}{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=2 is the signature of K11a86. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-10123456χ
15           11
13          2 -2
11         41 3
9        62  -4
7       74   3
5      76    -1
3     77     0
1    68      2
-1   36       -3
-3  26        4
-5 13         -2
-7 2          2
-91           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\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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K11a85

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K11a87