K11a229

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K11a228.gif

K11a228

K11a230.gif

K11a230

Contents

K11a229.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a229 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X20,6,21,5 X18,8,19,7 X14,10,15,9 X2,12,3,11 X10,14,11,13 X22,15,1,16 X8,18,9,17 X6,20,7,19 X16,21,17,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -9, 5, -7, 6, -2, 7, -5, 8, -11, 9, -4, 10, -3, 11, -8
Dowker-Thistlethwaite code 4 12 20 18 14 2 10 22 8 6 16
A Braid Representative
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A Morse Link Presentation K11a229 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a229/ThurstonBennequinNumber
Hyperbolic Volume 11.4893
A-Polynomial See Data:K11a229/A-polynomial

[edit Notes for K11a229'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 K11a229's four dimensional invariants]

Polynomial invariants

Alexander polynomial -4 t^2+18 t-27+18 t^{-1} -4 t^{-2}
Conway polynomial -4 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 71, 2 }
Jones polynomial -q^{10}+2 q^9-4 q^8+7 q^7-9 q^6+11 q^5-11 q^4+10 q^3-8 q^2+5 q-2+ q^{-1}
HOMFLY-PT polynomial (db, data sources) -z^4 a^{-2} -2 z^4 a^{-4} -z^4 a^{-6} -2 z^2 a^{-4} +z^2 a^{-6} +2 z^2 a^{-8} +z^2- a^{-4} + a^{-6} + a^{-8} - a^{-10} +1
Kauffman polynomial (db, data sources) z^{10} a^{-6} +z^{10} a^{-8} +2 z^9 a^{-5} +4 z^9 a^{-7} +2 z^9 a^{-9} +3 z^8 a^{-4} -z^8 a^{-8} +2 z^8 a^{-10} +4 z^7 a^{-3} -12 z^7 a^{-7} -7 z^7 a^{-9} +z^7 a^{-11} +3 z^6 a^{-2} -z^6 a^{-4} +2 z^6 a^{-6} -3 z^6 a^{-8} -9 z^6 a^{-10} +2 z^5 a^{-1} -6 z^5 a^{-3} -5 z^5 a^{-5} +14 z^5 a^{-7} +6 z^5 a^{-9} -5 z^5 a^{-11} -2 z^4 a^{-2} -5 z^4 a^{-4} -12 z^4 a^{-6} +2 z^4 a^{-8} +12 z^4 a^{-10} +z^4-2 z^3 a^{-1} +7 z^3 a^{-3} +5 z^3 a^{-5} -12 z^3 a^{-7} -z^3 a^{-9} +7 z^3 a^{-11} +7 z^2 a^{-4} +8 z^2 a^{-6} -2 z^2 a^{-8} -5 z^2 a^{-10} -2 z^2-2 z a^{-3} -2 z a^{-5} +2 z a^{-7} -2 z a^{-11} - a^{-4} - a^{-6} + a^{-8} + a^{-10} +1
The A2 invariant Data:K11a229/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a229/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a226,}

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

Vassiliev invariants

V2 and V3: (2, 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
8 56 32 \frac{844}{3} \frac{140}{3} 448 \frac{4400}{3} \frac{512}{3} 344 \frac{256}{3} 1568 \frac{6752}{3} \frac{1120}{3} \frac{121351}{15} -\frac{16724}{15} \frac{204964}{45} \frac{2921}{9} \frac{9511}{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 K11a229. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
21           1-1
19          1 1
17         31 -2
15        41  3
13       53   -2
11      64    2
9     55     0
7    56      -1
5   35       2
3  25        -3
1 14         3
-1 1          -1
-31           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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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K11a228

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K11a230