K11a338

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K11a337.gif

K11a337

K11a339.gif

K11a339

Contents

K11a338.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a338 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X14,4,15,3 X16,6,17,5 X20,8,21,7 X22,10,1,9 X18,12,19,11 X2,14,3,13 X4,16,5,15 X10,18,11,17 X12,20,13,19 X8,22,9,21
Gauss code 1, -7, 2, -8, 3, -1, 4, -11, 5, -9, 6, -10, 7, -2, 8, -3, 9, -6, 10, -4, 11, -5
Dowker-Thistlethwaite code 6 14 16 20 22 18 2 4 10 12 8
A Braid Representative
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A Morse Link Presentation K11a338 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 4
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a338/ThurstonBennequinNumber
Hyperbolic Volume 11.542
A-Polynomial See Data:K11a338/A-polynomial

[edit Notes for K11a338'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 K11a338's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^4-5 t^3+9 t^2-12 t+13-12 t^{-1} +9 t^{-2} -5 t^{-3} +2 t^{-4}
Conway polynomial 2 z^8+11 z^6+19 z^4+11 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 69, 8 }
Jones polynomial -q^{15}+3 q^{14}-6 q^{13}+8 q^{12}-10 q^{11}+11 q^{10}-10 q^9+8 q^8-6 q^7+4 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} +5 z^6 a^{-10} -z^6 a^{-12} +17 z^4 a^{-8} +6 z^4 a^{-10} -4 z^4 a^{-12} +16 z^2 a^{-8} -z^2 a^{-10} -4 z^2 a^{-12} +4 a^{-8} -2 a^{-10} - a^{-12}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +z^9 a^{-9} +4 z^9 a^{-11} +3 z^9 a^{-13} +z^8 a^{-8} -3 z^8 a^{-10} +2 z^8 a^{-12} +6 z^8 a^{-14} -4 z^7 a^{-9} -14 z^7 a^{-11} -2 z^7 a^{-13} +8 z^7 a^{-15} -7 z^6 a^{-8} -z^6 a^{-10} -13 z^6 a^{-12} -11 z^6 a^{-14} +8 z^6 a^{-16} +2 z^5 a^{-9} +8 z^5 a^{-11} -14 z^5 a^{-13} -14 z^5 a^{-15} +6 z^5 a^{-17} +17 z^4 a^{-8} +8 z^4 a^{-10} +6 z^4 a^{-12} +z^4 a^{-14} -11 z^4 a^{-16} +3 z^4 a^{-18} +6 z^3 a^{-9} +8 z^3 a^{-11} +13 z^3 a^{-13} +5 z^3 a^{-15} -5 z^3 a^{-17} +z^3 a^{-19} -16 z^2 a^{-8} -6 z^2 a^{-10} +5 z^2 a^{-12} -z^2 a^{-14} +4 z^2 a^{-16} -4 z a^{-9} -3 z a^{-11} -3 z a^{-13} -2 z a^{-15} +2 z a^{-17} +4 a^{-8} +2 a^{-10} - a^{-12}
The A2 invariant  q^{-14} +3 q^{-18} +2 q^{-22} - q^{-26} +2 q^{-28} -2 q^{-30} +2 q^{-32} -2 q^{-34} - q^{-36} - q^{-40} + q^{-42} - q^{-44}
The G2 invariant Data:K11a338/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: (11, 35)
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
44 280 968 \frac{6922}{3} \frac{974}{3} 12320 \frac{63568}{3} \frac{10912}{3} 2584 \frac{42592}{3} 39200 \frac{304568}{3} \frac{42856}{3} \frac{5975621}{30} \frac{119518}{15} \frac{3244882}{45} \frac{25243}{18} \frac{271301}{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=8 is the signature of K11a338. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
31           1-1
29          2 2
27         41 -3
25        42  2
23       64   -2
21      54    1
19     56     1
17    35      -2
15   35       2
13  13        -2
11  3         3
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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=8 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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K11a337.gif

K11a337

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K11a339