K11a20

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K11a19.gif

K11a19

K11a21.gif

K11a21

Contents

K11a20.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a20 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X12,6,13,5 X2837 X18,9,19,10 X6,12,7,11 X20,14,21,13 X22,16,1,15 X10,17,11,18 X16,20,17,19 X14,22,15,21
Gauss code 1, -4, 2, -1, 3, -6, 4, -2, 5, -9, 6, -3, 7, -11, 8, -10, 9, -5, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 8 12 2 18 6 20 22 10 16 14
A Braid Representative
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A Morse Link Presentation K11a20 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{2,3\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a20/ThurstonBennequinNumber
Hyperbolic Volume 15.3455
A-Polynomial See Data:K11a20/A-polynomial

[edit Notes for K11a20'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 K11a20's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^3+13 t^2-25 t+31-25 t^{-1} +13 t^{-2} -3 t^{-3}
Conway polynomial -3 z^6-5 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 113, 4 }
Jones polynomial -q^{11}+4 q^{10}-8 q^9+13 q^8-17 q^7+18 q^6-18 q^5+15 q^4-10 q^3+6 q^2-2 q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} -2 z^6 a^{-6} +z^4 a^{-2} -2 z^4 a^{-4} -7 z^4 a^{-6} +3 z^4 a^{-8} +3 z^2 a^{-2} -9 z^2 a^{-6} +7 z^2 a^{-8} -z^2 a^{-10} +2 a^{-2} + a^{-4} -5 a^{-6} +4 a^{-8} - a^{-10}
Kauffman polynomial (db, data sources) z^{10} a^{-6} +z^{10} a^{-8} +3 z^9 a^{-5} +7 z^9 a^{-7} +4 z^9 a^{-9} +3 z^8 a^{-4} +8 z^8 a^{-6} +12 z^8 a^{-8} +7 z^8 a^{-10} +2 z^7 a^{-3} -4 z^7 a^{-5} -8 z^7 a^{-7} +5 z^7 a^{-9} +7 z^7 a^{-11} +z^6 a^{-2} -6 z^6 a^{-4} -26 z^6 a^{-6} -30 z^6 a^{-8} -7 z^6 a^{-10} +4 z^6 a^{-12} -5 z^5 a^{-3} +z^5 a^{-5} -5 z^5 a^{-7} -23 z^5 a^{-9} -11 z^5 a^{-11} +z^5 a^{-13} -4 z^4 a^{-2} +3 z^4 a^{-4} +34 z^4 a^{-6} +30 z^4 a^{-8} -3 z^4 a^{-10} -6 z^4 a^{-12} +2 z^3 a^{-3} +14 z^3 a^{-7} +22 z^3 a^{-9} +5 z^3 a^{-11} -z^3 a^{-13} +5 z^2 a^{-2} -2 z^2 a^{-4} -22 z^2 a^{-6} -15 z^2 a^{-8} +2 z^2 a^{-10} +2 z^2 a^{-12} +z a^{-3} -z a^{-5} -7 z a^{-7} -7 z a^{-9} -2 z a^{-11} -2 a^{-2} + a^{-4} +5 a^{-6} +4 a^{-8} + a^{-10}
The A2 invariant Data:K11a20/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a20/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: (0, -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
0 -8 0 32 40 0 \frac{976}{3} \frac{352}{3} 152 0 32 0 0 1184 8 \frac{2264}{3} \frac{416}{3} 16

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 K11a20. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
23           1-1
21          3 3
19         51 -4
17        83  5
15       95   -4
13      98    1
11     99     0
9    69      -3
7   49       5
5  26        -4
3 15         4
1 1          -1
-11           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=5 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=9 {\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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K11a19

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K11a21