K11a363

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K11a362.gif

K11a362

K11a364.gif

K11a364

Contents

K11a363.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a363 at Knotilus!

K11a363 is also known as the pretzel knot P(5,1,5).



Knot presentations

Planar diagram presentation X8291 X16,4,17,3 X14,6,15,5 X18,8,19,7 X22,10,1,9 X20,12,21,11 X6,14,7,13 X4,16,5,15 X2,18,3,17 X12,20,13,19 X10,22,11,21
Gauss code 1, -9, 2, -8, 3, -7, 4, -1, 5, -11, 6, -10, 7, -3, 8, -2, 9, -4, 10, -6, 11, -5
Dowker-Thistlethwaite code 8 16 14 18 22 20 6 4 2 12 10
A Braid Representative
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A Morse Link Presentation K11a363 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 1
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a363/ThurstonBennequinNumber
Hyperbolic Volume 6.29702
A-Polynomial See Data:K11a363/A-polynomial

[edit Notes for K11a363's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 1
Rasmussen s-Invariant -2

[edit Notes for K11a363's four dimensional invariants]

Polynomial invariants

Alexander polynomial 9 t-17+9 t^{-1}
Conway polynomial 9 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 35, 2 }
Jones polynomial -q^{12}+q^{11}-2 q^{10}+3 q^9-4 q^8+5 q^7-4 q^6+5 q^5-4 q^4+3 q^3-2 q^2+q
HOMFLY-PT polynomial (db, data sources) z^2 a^{-2} +2 z^2 a^{-4} +3 z^2 a^{-6} +2 z^2 a^{-8} +z^2 a^{-10} +2 a^{-6} - 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} -4 z^8 a^{-10} -7 z^8 a^{-12} +4 z^7 a^{-7} -8 z^7 a^{-9} -20 z^7 a^{-11} -8 z^7 a^{-13} +5 z^6 a^{-6} -10 z^6 a^{-8} +z^6 a^{-10} +16 z^6 a^{-12} +4 z^5 a^{-5} -12 z^5 a^{-7} +6 z^5 a^{-9} +44 z^5 a^{-11} +22 z^5 a^{-13} +3 z^4 a^{-4} -14 z^4 a^{-6} +4 z^4 a^{-8} +8 z^4 a^{-10} -13 z^4 a^{-12} +2 z^3 a^{-3} -6 z^3 a^{-5} +6 z^3 a^{-7} +2 z^3 a^{-9} -36 z^3 a^{-11} -24 z^3 a^{-13} +z^2 a^{-2} -2 z^2 a^{-4} +9 z^2 a^{-6} +4 z^2 a^{-8} -5 z^2 a^{-10} +3 z^2 a^{-12} +9 z a^{-11} +9 z a^{-13} -2 a^{-6} - a^{-12}
The A2 invariant Data:K11a363/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a363/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, 27)
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
36 216 648 1722 318 7776 14832 2592 2520 7776 23328 61992 11448 \frac{1284333}{10} -\frac{18546}{5} \frac{304022}{5} \frac{2449}{2} \frac{87853}{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=2 is the signature of K11a363. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
25           1-1
23            0
21         21 -1
19        1   1
17       32   -1
15      21    1
13     23     1
11    32      1
9   12       1
7  23        -1
5  1         1
312          -1
11           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}^{2}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11a362.gif

K11a362

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K11a364