K11n17

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K11n16.gif

K11n16

K11n18.gif

K11n18

Contents

K11n17.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n17 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X7,16,8,17 X2,9,3,10 X18,11,19,12 X20,13,21,14 X15,6,16,7 X22,17,1,18 X14,19,15,20 X12,21,13,22
Gauss code 1, -5, 2, -1, 3, 8, -4, -2, 5, -3, 6, -11, 7, -10, -8, 4, 9, -6, 10, -7, 11, -9
Dowker-Thistlethwaite code 4 8 10 -16 2 18 20 -6 22 14 12
A Braid Representative
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A Morse Link Presentation K11n17 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n17/ThurstonBennequinNumber
Hyperbolic Volume 11.3588
A-Polynomial See Data:K11n17/A-polynomial

[edit Notes for K11n17's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n17's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^2+12 t-19+12 t^{-1} -2 t^{-2}
Conway polynomial -2 z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 47, -2 }
Jones polynomial 2 q^{-1} -4 q^{-2} +6 q^{-3} -7 q^{-4} +8 q^{-5} -7 q^{-6} +6 q^{-7} -4 q^{-8} +2 q^{-9} - q^{-10}
HOMFLY-PT polynomial (db, data sources) -a^{10}+2 z^2 a^8+a^8-z^4 a^6-z^4 a^4+2 z^2 a^2+a^2
Kauffman polynomial (db, data sources) z^7 a^{11}-5 z^5 a^{11}+8 z^3 a^{11}-4 z a^{11}+2 z^8 a^{10}-9 z^6 a^{10}+12 z^4 a^{10}-5 z^2 a^{10}+a^{10}+z^9 a^9-12 z^5 a^9+17 z^3 a^9-5 z a^9+5 z^8 a^8-17 z^6 a^8+14 z^4 a^8-4 z^2 a^8+a^8+z^9 a^7+3 z^7 a^7-16 z^5 a^7+12 z^3 a^7-3 z a^7+3 z^8 a^6-5 z^6 a^6-z^4 a^6+4 z^7 a^5-8 z^5 a^5+6 z^3 a^5-3 z a^5+3 z^6 a^4-3 z^4 a^4+2 z^2 a^4+z^5 a^3+3 z^3 a^3-z a^3+3 z^2 a^2-a^2
The A2 invariant Data:K11n17/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n17/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (4, -10)
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
16 -80 128 \frac{1352}{3} \frac{232}{3} -1280 -\frac{8288}{3} -\frac{1472}{3} -432 \frac{2048}{3} 3200 \frac{21632}{3} \frac{3712}{3} \frac{257702}{15} -\frac{2168}{15} \frac{344648}{45} \frac{1354}{9} \frac{15542}{15}

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 K11n17. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-9-8-7-6-5-4-3-2-10χ
-1         22
-3        31-2
-5       31 2
-7      43  -1
-9     43   1
-11    34    1
-13   34     -1
-15  13      2
-17 13       -2
-19 1        1
-211         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11n16.gif

K11n16

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K11n18