K11n109

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K11n108.gif

K11n108

K11n110.gif

K11n110

Contents

K11n109.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n109 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X5,14,6,15 X16,7,17,8 X18,9,19,10 X2,11,3,12 X20,13,21,14 X15,22,16,1 X8,17,9,18 X12,19,13,20 X21,7,22,6
Gauss code 1, -6, 2, -1, -3, 11, 4, -9, 5, -2, 6, -10, 7, 3, -8, -4, 9, -5, 10, -7, -11, 8
Dowker-Thistlethwaite code 4 10 -14 16 18 2 20 -22 8 12 -6
A Braid Representative
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A Morse Link Presentation K11n109 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n109/ThurstonBennequinNumber
Hyperbolic Volume 13.0869
A-Polynomial See Data:K11n109/A-polynomial

[edit Notes for K11n109's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 4

[edit Notes for K11n109's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+7 t^2-13 t+15-13 t^{-1} +7 t^{-2} - t^{-3}
Conway polynomial -z^6+z^4+6 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 57, -4 }
Jones polynomial 2 q^{-2} -4 q^{-3} +7 q^{-4} -9 q^{-5} +10 q^{-6} -9 q^{-7} +8 q^{-8} -5 q^{-9} +2 q^{-10} - q^{-11}
HOMFLY-PT polynomial (db, data sources) -z^2 a^{10}-2 a^{10}+2 z^4 a^8+5 z^2 a^8+3 a^8-z^6 a^6-3 z^4 a^6-3 z^2 a^6-2 a^6+2 z^4 a^4+5 z^2 a^4+2 a^4
Kauffman polynomial (db, data sources) z^5 a^{13}-3 z^3 a^{13}+2 z a^{13}+2 z^6 a^{12}-4 z^4 a^{12}+z^2 a^{12}+3 z^7 a^{11}-6 z^5 a^{11}+4 z^3 a^{11}-2 z a^{11}+3 z^8 a^{10}-7 z^6 a^{10}+11 z^4 a^{10}-10 z^2 a^{10}+2 a^{10}+z^9 a^9+3 z^7 a^9-10 z^5 a^9+12 z^3 a^9-4 z a^9+5 z^8 a^8-13 z^6 a^8+20 z^4 a^8-13 z^2 a^8+3 a^8+z^9 a^7+z^7 a^7-2 z^5 a^7+2 z^3 a^7+2 z^8 a^6-4 z^6 a^6+8 z^4 a^6-8 z^2 a^6+2 a^6+z^7 a^5+z^5 a^5-3 z^3 a^5+3 z^4 a^4-6 z^2 a^4+2 a^4
The A2 invariant -q^{34}-q^{32}-2 q^{28}+2 q^{26}+q^{24}+2 q^{20}-2 q^{18}+2 q^{16}-q^{14}+2 q^{10}-q^8+2 q^6
The G2 invariant Data:K11n109/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n137,}

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

Vassiliev invariants

V2 and V3: (6, -16)
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
24 -128 288 828 132 -3072 -\frac{17792}{3} -\frac{3104}{3} -864 2304 8192 19872 3168 \frac{215311}{5} \frac{1476}{5} \frac{269564}{15} \frac{1201}{3} \frac{11951}{5}

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 K11n109. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-9-8-7-6-5-4-3-2-10χ
-3         22
-5        31-2
-7       41 3
-9      53  -2
-11     54   1
-13    45    1
-15   45     -1
-17  14      3
-19 14       -3
-21 1        1
-231         -1
Integral Khovanov Homology

(db, data source)

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

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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K11n108.gif

K11n108

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K11n110