K11a204

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K11a203.gif

K11a203

K11a205.gif

K11a205

Contents

K11a204.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a204 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X16,6,17,5 X18,8,19,7 X20,10,21,9 X2,12,3,11 X22,13,1,14 X10,16,11,15 X8,18,9,17 X6,20,7,19 X14,21,15,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -9, 5, -8, 6, -2, 7, -11, 8, -3, 9, -4, 10, -5, 11, -7
Dowker-Thistlethwaite code 4 12 16 18 20 2 22 10 8 6 14
A Braid Representative
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A Morse Link Presentation K11a204 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a204/ThurstonBennequinNumber
Hyperbolic Volume 13.608
A-Polynomial See Data:K11a204/A-polynomial

[edit Notes for K11a204'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 K11a204's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+11 t^2-23 t+29-23 t^{-1} +11 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 101, 4 }
Jones polynomial -q^{11}+3 q^{10}-6 q^9+10 q^8-14 q^7+16 q^6-16 q^5+14 q^4-10 q^3+7 q^2-3 q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} -z^6 a^{-6} +z^4 a^{-2} -2 z^4 a^{-4} -2 z^4 a^{-6} +2 z^4 a^{-8} +2 z^2 a^{-2} -2 z^2 a^{-6} +4 z^2 a^{-8} -z^2 a^{-10} + a^{-2} + a^{-4} -2 a^{-6} +2 a^{-8} - a^{-10}
Kauffman polynomial (db, data sources) z^{10} a^{-6} +z^{10} a^{-8} +3 z^9 a^{-5} +6 z^9 a^{-7} +3 z^9 a^{-9} +4 z^8 a^{-4} +6 z^8 a^{-6} +7 z^8 a^{-8} +5 z^8 a^{-10} +3 z^7 a^{-3} -2 z^7 a^{-5} -9 z^7 a^{-7} +z^7 a^{-9} +5 z^7 a^{-11} +z^6 a^{-2} -8 z^6 a^{-4} -16 z^6 a^{-6} -19 z^6 a^{-8} -9 z^6 a^{-10} +3 z^6 a^{-12} -8 z^5 a^{-3} -6 z^5 a^{-5} +3 z^5 a^{-7} -11 z^5 a^{-9} -11 z^5 a^{-11} +z^5 a^{-13} -3 z^4 a^{-2} +z^4 a^{-4} +12 z^4 a^{-6} +22 z^4 a^{-8} +8 z^4 a^{-10} -6 z^4 a^{-12} +5 z^3 a^{-3} +3 z^3 a^{-5} +2 z^3 a^{-7} +15 z^3 a^{-9} +9 z^3 a^{-11} -2 z^3 a^{-13} +3 z^2 a^{-2} +z^2 a^{-4} -7 z^2 a^{-6} -10 z^2 a^{-8} -3 z^2 a^{-10} +2 z^2 a^{-12} -z a^{-5} -2 z a^{-7} -4 z a^{-9} -3 z a^{-11} - a^{-2} + a^{-4} +2 a^{-6} +2 a^{-8} + a^{-10}
The A2 invariant Data:K11a204/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a204/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a222,}

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

Vassiliev invariants

V2 and V3: (3, 7)
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
12 56 72 286 42 672 \frac{4688}{3} \frac{608}{3} 280 288 1568 3432 504 \frac{87151}{10} -\frac{8386}{15} \frac{60502}{15} \frac{1489}{6} \frac{5391}{10}

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 K11a204. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
23           1-1
21          2 2
19         41 -3
17        62  4
15       84   -4
13      86    2
11     88     0
9    68      -2
7   48       4
5  36        -3
3 15         4
1 2          -2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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K11a203.gif

K11a203

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K11a205