K11a17

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K11a16.gif

K11a16

K11a18.gif

K11a18

Contents

K11a17.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a17 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X12,6,13,5 X2837 X16,9,17,10 X6,12,7,11 X20,14,21,13 X10,15,11,16 X22,17,1,18 X14,20,15,19 X18,21,19,22
Gauss code 1, -4, 2, -1, 3, -6, 4, -2, 5, -8, 6, -3, 7, -10, 8, -5, 9, -11, 10, -7, 11, -9
Dowker-Thistlethwaite code 4 8 12 2 16 6 20 10 22 14 18
A Braid Representative
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A Morse Link Presentation K11a17 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a17/ThurstonBennequinNumber
Hyperbolic Volume 16.1013
A-Polynomial See Data:K11a17/A-polynomial

[edit Notes for K11a17's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -2

[edit Notes for K11a17's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-10 t^2+30 t-41+30 t^{-1} -10 t^{-2} + t^{-3}
Conway polynomial z^6-4 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 123, 2 }
Jones polynomial -q^8+3 q^7-7 q^6+13 q^5-17 q^4+20 q^3-20 q^2+17 q-13+8 q^{-1} -3 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^4 a^{-2} -3 z^4 a^{-4} -2 z^4+a^2 z^2-3 z^2 a^{-4} +3 z^2 a^{-6} -2 z^2+a^2- a^{-2} +2 a^{-6} - a^{-8}
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +4 z^9 a^{-1} +8 z^9 a^{-3} +4 z^9 a^{-5} +13 z^8 a^{-2} +14 z^8 a^{-4} +6 z^8 a^{-6} +5 z^8+3 a z^7-z^7 a^{-1} -5 z^7 a^{-3} +4 z^7 a^{-5} +5 z^7 a^{-7} +a^2 z^6-34 z^6 a^{-2} -32 z^6 a^{-4} -6 z^6 a^{-6} +3 z^6 a^{-8} -10 z^6-7 a z^5-11 z^5 a^{-1} -13 z^5 a^{-3} -16 z^5 a^{-5} -6 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+28 z^4 a^{-2} +26 z^4 a^{-4} +z^4 a^{-6} -5 z^4 a^{-8} +5 z^4+5 a z^3+6 z^3 a^{-1} +10 z^3 a^{-3} +13 z^3 a^{-5} +2 z^3 a^{-7} -2 z^3 a^{-9} +3 a^2 z^2-12 z^2 a^{-2} -8 z^2 a^{-4} +3 z^2 a^{-6} +3 z^2 a^{-8} -z^2-a z-2 z a^{-5} +z a^{-9} -a^2+ a^{-2} -2 a^{-6} - a^{-8}
The A2 invariant Data:K11a17/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a17/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: (-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{322}{3} \frac{110}{3} -64 \frac{448}{3} \frac{64}{3} 48 -\frac{32}{3} 128 -\frac{1288}{3} -\frac{440}{3} \frac{3089}{30} -\frac{418}{15} -\frac{2222}{45} \frac{1615}{18} -\frac{1231}{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 K11a17. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
17           1-1
15          2 2
13         51 -4
11        82  6
9       95   -4
7      118    3
5     99     0
3    811      -3
1   610       4
-1  27        -5
-3 16         5
-5 2          -2
-71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\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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K11a16.gif

K11a16

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K11a18