K11a364

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K11a363.gif

K11a363

K11a365.gif

K11a365

Contents

K11a364.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a364 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 X6,18,7,17 X8,20,9,19 X12,22,13,21
Gauss code 1, -7, 2, -8, 3, -9, 4, -10, 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 6 8 12
A Braid Representative
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A Morse Link Presentation K11a364 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 4
3-genus 4
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a364/ThurstonBennequinNumber
Hyperbolic Volume 5.14021
A-Polynomial See Data:K11a364/A-polynomial

[edit Notes for K11a364's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant -8

[edit Notes for K11a364's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^4-3 t^3+3 t^2-3 t+3-3 t^{-1} +3 t^{-2} -3 t^{-3} +2 t^{-4}
Conway polynomial 2 z^8+13 z^6+25 z^4+14 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 25, 8 }
Jones polynomial -q^{15}+q^{14}-2 q^{13}+3 q^{12}-3 q^{11}+3 q^{10}-3 q^9+3 q^8-2 q^7+2 q^6-q^5+q^4
HOMFLY-PT polynomial (db, data sources) z^8 a^{-8} +z^8 a^{-10} +7 z^6 a^{-8} +7 z^6 a^{-10} -z^6 a^{-12} +15 z^4 a^{-8} +16 z^4 a^{-10} -6 z^4 a^{-12} +10 z^2 a^{-8} +14 z^2 a^{-10} -10 z^2 a^{-12} + a^{-8} +4 a^{-10} -4 a^{-12}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +z^9 a^{-9} +2 z^9 a^{-11} +z^9 a^{-13} +z^8 a^{-8} -7 z^8 a^{-10} -7 z^8 a^{-12} +z^8 a^{-14} -6 z^7 a^{-9} -12 z^7 a^{-11} -5 z^7 a^{-13} +z^7 a^{-15} -7 z^6 a^{-8} +18 z^6 a^{-10} +20 z^6 a^{-12} -4 z^6 a^{-14} +z^6 a^{-16} +10 z^5 a^{-9} +23 z^5 a^{-11} +9 z^5 a^{-13} -3 z^5 a^{-15} +z^5 a^{-17} +15 z^4 a^{-8} -24 z^4 a^{-10} -30 z^4 a^{-12} +6 z^4 a^{-14} -2 z^4 a^{-16} +z^4 a^{-18} -4 z^3 a^{-9} -18 z^3 a^{-11} -8 z^3 a^{-13} +4 z^3 a^{-15} -z^3 a^{-17} +z^3 a^{-19} -10 z^2 a^{-8} +17 z^2 a^{-10} +19 z^2 a^{-12} -4 z^2 a^{-14} +3 z^2 a^{-16} -z^2 a^{-18} +5 z a^{-11} +z a^{-13} -z a^{-15} +z a^{-17} -2 z a^{-19} + a^{-8} -4 a^{-10} -4 a^{-12}
The A2 invariant  q^{-14} + q^{-18} + q^{-22} + q^{-24} + q^{-26} + q^{-28} + q^{-32} - q^{-38} - q^{-40} - q^{-42} - q^{-44}
The G2 invariant Data:K11a364/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: (14, 50)
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
56 400 1568 \frac{11284}{3} \frac{1724}{3} 22400 \frac{117184}{3} \frac{20896}{3} 5008 \frac{87808}{3} 80000 \frac{631904}{3} \frac{96544}{3} \frac{6205057}{15} \frac{214012}{15} \frac{7077268}{45} \frac{20495}{9} \frac{305617}{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=8 is the signature of K11a364. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
31           1-1
29            0
27         21 -1
25        1   1
23       22   0
21      11    0
19     22     0
17    11      0
15   12       1
13  11        0
11  1         1
911          0
71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=7 i=9
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=9 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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K11a363.gif

K11a363

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K11a365