K11a365

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K11a364.gif

K11a364

K11a366.gif

K11a366

Contents

K11a365.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a365 at Knotilus!



Knot presentations

Planar diagram presentation X10,2,11,1 X14,4,15,3 X16,6,17,5 X18,8,19,7 X20,10,21,9 X22,12,1,11 X2,14,3,13 X4,16,5,15 X8,18,9,17 X6,20,7,19 X12,22,13,21
Gauss code 1, -7, 2, -8, 3, -10, 4, -9, 5, -1, 6, -11, 7, -2, 8, -3, 9, -4, 10, -5, 11, -6
Dowker-Thistlethwaite code 10 14 16 18 20 22 2 4 8 6 12
A Braid Representative
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A Morse Link Presentation K11a365 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{3,4\}
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a365/ThurstonBennequinNumber
Hyperbolic Volume 9.83268
A-Polynomial See Data:K11a365/A-polynomial

[edit Notes for K11a365'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 K11a365's four dimensional invariants]

Polynomial invariants

Alexander polynomial 4 t^3-8 t^2+9 t-9+9 t^{-1} -8 t^{-2} +4 t^{-3}
Conway polynomial 4 z^6+16 z^4+13 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 51, 6 }
Jones polynomial -q^{14}+q^{13}-3 q^{12}+5 q^{11}-6 q^{10}+8 q^9-8 q^8+7 q^7-5 q^6+4 q^5-2 q^4+q^3
HOMFLY-PT polynomial (db, data sources) z^6 a^{-6} +2 z^6 a^{-8} +z^6 a^{-10} +4 z^4 a^{-6} +9 z^4 a^{-8} +4 z^4 a^{-10} -z^4 a^{-12} +3 z^2 a^{-6} +10 z^2 a^{-8} +4 z^2 a^{-10} -4 z^2 a^{-12} +2 a^{-8} +2 a^{-10} -3 a^{-12}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +2 z^9 a^{-9} +3 z^9 a^{-11} +z^9 a^{-13} +3 z^8 a^{-8} -3 z^8 a^{-10} -5 z^8 a^{-12} +z^8 a^{-14} +2 z^7 a^{-7} -7 z^7 a^{-9} -13 z^7 a^{-11} -3 z^7 a^{-13} +z^7 a^{-15} +z^6 a^{-6} -13 z^6 a^{-8} +4 z^6 a^{-10} +15 z^6 a^{-12} -2 z^6 a^{-14} +z^6 a^{-16} -7 z^5 a^{-7} +7 z^5 a^{-9} +23 z^5 a^{-11} +7 z^5 a^{-13} -z^5 a^{-15} +z^5 a^{-17} -4 z^4 a^{-6} +18 z^4 a^{-8} -5 z^4 a^{-10} -20 z^4 a^{-12} +5 z^4 a^{-14} -2 z^4 a^{-16} +4 z^3 a^{-7} -3 z^3 a^{-9} -15 z^3 a^{-11} -5 z^3 a^{-13} -z^3 a^{-15} -4 z^3 a^{-17} +3 z^2 a^{-6} -12 z^2 a^{-8} +6 z^2 a^{-10} +15 z^2 a^{-12} -6 z^2 a^{-14} +5 z a^{-11} +z a^{-13} +4 z a^{-17} +2 a^{-8} -2 a^{-10} -3 a^{-12}
The A2 invariant Data:K11a365/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a365/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: (13, 45)
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
52 360 1352 \frac{9926}{3} \frac{1618}{3} 18720 33360 5984 4648 \frac{70304}{3} 64800 \frac{516152}{3} \frac{84136}{3} \frac{10246243}{30} \frac{94274}{15} \frac{6248846}{45} \frac{37565}{18} \frac{558883}{30}

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 K11a365. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
29           1-1
27            0
25         31 -2
23        2   2
21       43   -1
19      42    2
17     44     0
15    34      -1
13   24       2
11  23        -1
9  2         2
712          -1
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}^{2}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=8 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=9 {\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=10 {\mathbb Z}
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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K11a364.gif

K11a364

K11a366.gif

K11a366