K11a291

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K11a290.gif

K11a290

K11a292.gif

K11a292

Contents

K11a291.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a291 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,4,11,3 X16,6,17,5 X22,8,1,7 X4,10,5,9 X18,12,19,11 X20,14,21,13 X2,16,3,15 X8,18,9,17 X12,20,13,19 X14,22,15,21
Gauss code 1, -8, 2, -5, 3, -1, 4, -9, 5, -2, 6, -10, 7, -11, 8, -3, 9, -6, 10, -7, 11, -4
Dowker-Thistlethwaite code 6 10 16 22 4 18 20 2 8 12 14
A Braid Representative
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A Morse Link Presentation K11a291 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{3,4\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a291/ThurstonBennequinNumber
Hyperbolic Volume 14.5799
A-Polynomial See Data:K11a291/A-polynomial

[edit Notes for K11a291's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -6

[edit Notes for K11a291's four dimensional invariants]

Polynomial invariants

Alexander polynomial 5 t^3-14 t^2+20 t-21+20 t^{-1} -14 t^{-2} +5 t^{-3}
Conway polynomial 5 z^6+16 z^4+9 z^2+1
2nd Alexander ideal (db, data sources) \{3,t+1\}
Determinant and Signature { 99, 6 }
Jones polynomial -q^{14}+3 q^{13}-6 q^{12}+10 q^{11}-14 q^{10}+15 q^9-16 q^8+14 q^7-9 q^6+7 q^5-3 q^4+q^3
HOMFLY-PT polynomial (db, data sources) z^6 a^{-6} +3 z^6 a^{-8} +z^6 a^{-10} +3 z^4 a^{-6} +13 z^4 a^{-8} +z^4 a^{-10} -z^4 a^{-12} +z^2 a^{-6} +16 z^2 a^{-8} -6 z^2 a^{-10} -2 z^2 a^{-12} +6 a^{-8} -6 a^{-10} + a^{-12}
Kauffman polynomial (db, data sources) 2 z^{10} a^{-10} +2 z^{10} a^{-12} +5 z^9 a^{-9} +9 z^9 a^{-11} +4 z^9 a^{-13} +6 z^8 a^{-8} +2 z^8 a^{-10} +4 z^8 a^{-14} +3 z^7 a^{-7} -14 z^7 a^{-9} -29 z^7 a^{-11} -8 z^7 a^{-13} +4 z^7 a^{-15} +z^6 a^{-6} -22 z^6 a^{-8} -22 z^6 a^{-10} -5 z^6 a^{-12} -3 z^6 a^{-14} +3 z^6 a^{-16} -8 z^5 a^{-7} +8 z^5 a^{-9} +36 z^5 a^{-11} +14 z^5 a^{-13} -5 z^5 a^{-15} +z^5 a^{-17} -3 z^4 a^{-6} +27 z^4 a^{-8} +35 z^4 a^{-10} +8 z^4 a^{-12} -3 z^4 a^{-14} -6 z^4 a^{-16} +2 z^3 a^{-7} +6 z^3 a^{-9} -10 z^3 a^{-11} -13 z^3 a^{-13} -z^3 a^{-15} -2 z^3 a^{-17} +z^2 a^{-6} -18 z^2 a^{-8} -19 z^2 a^{-10} -2 z^2 a^{-12} +z^2 a^{-14} +3 z^2 a^{-16} -6 z a^{-9} -4 z a^{-11} +2 z a^{-13} +z a^{-15} +z a^{-17} +6 a^{-8} +6 a^{-10} + a^{-12}
The A2 invariant Data:K11a291/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a291/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: (9, 25)
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
36 200 648 1450 190 7200 \frac{35216}{3} \frac{5888}{3} 1288 7776 20000 52200 6840 \frac{980253}{10} \frac{85162}{15} \frac{481426}{15} \frac{4003}{6} \frac{38653}{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=6 is the signature of K11a291. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
29           1-1
27          2 2
25         41 -3
23        62  4
21       84   -4
19      76    1
17     98     -1
15    57      -2
13   49       5
11  35        -2
9  4         4
713          -2
51           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=5 i=7
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}^{3}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=6 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=7 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=8 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=9 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=10 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=11 {\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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K11a290

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K11a292