K11a18

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K11a17.gif

K11a17

K11a19.gif

K11a19

Contents

K11a18.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a18 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X12,6,13,5 X2837 X16,10,17,9 X6,12,7,11 X20,13,21,14 X18,16,19,15 X10,18,11,17 X22,19,1,20 X14,21,15,22
Gauss code 1, -4, 2, -1, 3, -6, 4, -2, 5, -9, 6, -3, 7, -11, 8, -5, 9, -8, 10, -7, 11, -10
Dowker-Thistlethwaite code 4 8 12 2 16 6 20 18 10 22 14
A Braid Representative
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A Morse Link Presentation K11a18 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2,3\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a18/ThurstonBennequinNumber
Hyperbolic Volume 16.03
A-Polynomial See Data:K11a18/A-polynomial

[edit Notes for K11a18'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 K11a18's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^3-13 t^2+29 t-37+29 t^{-1} -13 t^{-2} +3 t^{-3}
Conway polynomial 3 z^6+5 z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 127, 2 }
Jones polynomial q^9-4 q^8+8 q^7-14 q^6+18 q^5-20 q^4+21 q^3-17 q^2+13 q-7+3 q^{-1} - q^{-2}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +2 z^6 a^{-4} +2 z^4 a^{-2} +7 z^4 a^{-4} -3 z^4 a^{-6} -z^4+2 z^2 a^{-2} +10 z^2 a^{-4} -7 z^2 a^{-6} +z^2 a^{-8} -2 z^2+ a^{-2} +5 a^{-4} -5 a^{-6} + a^{-8} -1
Kauffman polynomial (db, data sources) z^{10} a^{-4} +z^{10} a^{-6} +4 z^9 a^{-3} +8 z^9 a^{-5} +4 z^9 a^{-7} +6 z^8 a^{-2} +15 z^8 a^{-4} +15 z^8 a^{-6} +6 z^8 a^{-8} +5 z^7 a^{-1} +5 z^7 a^{-3} +4 z^7 a^{-7} +4 z^7 a^{-9} -6 z^6 a^{-2} -35 z^6 a^{-4} -39 z^6 a^{-6} -12 z^6 a^{-8} +z^6 a^{-10} +3 z^6+a z^5-6 z^5 a^{-1} -20 z^5 a^{-3} -33 z^5 a^{-5} -30 z^5 a^{-7} -10 z^5 a^{-9} +2 z^4 a^{-2} +31 z^4 a^{-4} +31 z^4 a^{-6} +5 z^4 a^{-8} -2 z^4 a^{-10} -5 z^4-2 a z^3+2 z^3 a^{-1} +20 z^3 a^{-3} +38 z^3 a^{-5} +30 z^3 a^{-7} +8 z^3 a^{-9} +z^2 a^{-2} -15 z^2 a^{-4} -15 z^2 a^{-6} -z^2 a^{-8} +z^2 a^{-10} +3 z^2+a z-6 z a^{-3} -14 z a^{-5} -11 z a^{-7} -2 z a^{-9} - a^{-2} +5 a^{-4} +5 a^{-6} + a^{-8} -1
The A2 invariant Data:K11a18/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a18/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, 5)
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 40 128 \frac{584}{3} \frac{64}{3} 640 \frac{2704}{3} \frac{448}{3} 104 \frac{2048}{3} 800 \frac{9344}{3} \frac{1024}{3} \frac{69302}{15} \frac{4432}{15} \frac{66128}{45} \frac{298}{9} \frac{2342}{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 K11a18. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-1012345678χ
19           11
17          3 -3
15         51 4
13        93  -6
11       95   4
9      119    -2
7     109     1
5    711      4
3   610       -4
1  28        6
-1 15         -4
-3 2          2
-51           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\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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K11a17

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K11a19