K11a106

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K11a105.gif

K11a105

K11a107.gif

K11a107

Contents

K11a106.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a106 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,5,15,6 X16,7,17,8 X2,10,3,9 X20,12,21,11 X18,13,19,14 X6,15,7,16 X8,17,9,18 X22,20,1,19 X12,22,13,21
Gauss code 1, -5, 2, -1, 3, -8, 4, -9, 5, -2, 6, -11, 7, -3, 8, -4, 9, -7, 10, -6, 11, -10
Dowker-Thistlethwaite code 4 10 14 16 2 20 18 6 8 22 12
A Braid Representative
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A Morse Link Presentation K11a106 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:K11a106/ThurstonBennequinNumber
Hyperbolic Volume 13.4001
A-Polynomial See Data:K11a106/A-polynomial

[edit Notes for K11a106'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 K11a106's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+12 t^2-18 t+21-18 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 { 93, 0 }
Jones polynomial q^6-3 q^5+5 q^4-9 q^3+13 q^2-14 q+15-13 q^{-1} +10 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) z^8-a^2 z^6-2 z^6 a^{-2} +6 z^6-4 a^2 z^4-9 z^4 a^{-2} +z^4 a^{-4} +14 z^4-5 a^2 z^2-12 z^2 a^{-2} +3 z^2 a^{-4} +15 z^2-2 a^2-4 a^{-2} + a^{-4} +6
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10}+3 a z^9+6 z^9 a^{-1} +3 z^9 a^{-3} +4 a^2 z^8+6 z^8 a^{-2} +4 z^8 a^{-4} +6 z^8+4 a^3 z^7-2 a z^7-13 z^7 a^{-1} -4 z^7 a^{-3} +3 z^7 a^{-5} +3 a^4 z^6-4 a^2 z^6-27 z^6 a^{-2} -12 z^6 a^{-4} +z^6 a^{-6} -21 z^6+a^5 z^5-5 a^3 z^5+7 z^5 a^{-1} -9 z^5 a^{-3} -10 z^5 a^{-5} -6 a^4 z^4+a^2 z^4+39 z^4 a^{-2} +10 z^4 a^{-4} -3 z^4 a^{-6} +33 z^4-2 a^5 z^3-a^3 z^3+a z^3+9 z^3 a^{-1} +17 z^3 a^{-3} +8 z^3 a^{-5} +3 a^4 z^2-4 a^2 z^2-22 z^2 a^{-2} -5 z^2 a^{-4} +z^2 a^{-6} -23 z^2+a^5 z+a^3 z-2 a z-6 z a^{-1} -6 z a^{-3} -2 z a^{-5} +2 a^2+4 a^{-2} + a^{-4} +6
The A2 invariant -q^{14}+q^{12}-2 q^{10}+q^8+q^6-q^4+4 q^2-2+3 q^{-2} +2 q^{-8} -3 q^{-10} - q^{-14} + q^{-18}
The G2 invariant q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+8 q^{72}-6 q^{70}-2 q^{68}+16 q^{66}-29 q^{64}+41 q^{62}-42 q^{60}+27 q^{58}-2 q^{56}-37 q^{54}+74 q^{52}-101 q^{50}+102 q^{48}-79 q^{46}+25 q^{44}+47 q^{42}-117 q^{40}+170 q^{38}-174 q^{36}+125 q^{34}-38 q^{32}-72 q^{30}+154 q^{28}-185 q^{26}+155 q^{24}-59 q^{22}-46 q^{20}+126 q^{18}-135 q^{16}+69 q^{14}+37 q^{12}-137 q^{10}+180 q^8-143 q^6+34 q^4+117 q^2-236+292 q^{-2} -242 q^{-4} +106 q^{-6} +61 q^{-8} -211 q^{-10} +291 q^{-12} -270 q^{-14} +173 q^{-16} -22 q^{-18} -114 q^{-20} +200 q^{-22} -191 q^{-24} +99 q^{-26} +17 q^{-28} -118 q^{-30} +151 q^{-32} -107 q^{-34} + q^{-36} +114 q^{-38} -186 q^{-40} +195 q^{-42} -128 q^{-44} -5 q^{-46} +121 q^{-48} -198 q^{-50} +207 q^{-52} -152 q^{-54} +59 q^{-56} +37 q^{-58} -105 q^{-60} +135 q^{-62} -120 q^{-64} +77 q^{-66} -23 q^{-68} -18 q^{-70} +42 q^{-72} -48 q^{-74} +41 q^{-76} -25 q^{-78} +11 q^{-80} +2 q^{-82} -8 q^{-84} +7 q^{-86} -7 q^{-88} +4 q^{-90} -2 q^{-92} + q^{-94}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a194, K11a346,}

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

Vassiliev invariants

V2 and V3: (1, -1)
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 -8 8 -\frac{34}{3} -\frac{38}{3} -32 \frac{16}{3} \frac{64}{3} 24 \frac{32}{3} 32 -\frac{136}{3} -\frac{152}{3} -\frac{209}{30} -\frac{662}{15} \frac{1862}{45} -\frac{463}{18} \frac{751}{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 K11a106. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-10123456χ
13           11
11          2 -2
9         31 2
7        62  -4
5       73   4
3      76    -1
1     87     1
-1    68      2
-3   47       -3
-5  26        4
-7 14         -3
-9 2          2
-111           -1
Integral Khovanov Homology

(db, data source)

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

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K11a105.gif

K11a105

K11a107.gif

K11a107