K11a361

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K11a360.gif

K11a360

K11a362.gif

K11a362

Contents

K11a361.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a361 at Knotilus!



Knot presentations

Planar diagram presentation X8291 X14,4,15,3 X18,6,19,5 X16,8,17,7 X20,10,21,9 X22,12,1,11 X4,14,5,13 X2,16,3,15 X6,18,7,17 X12,20,13,19 X10,22,11,21
Gauss code 1, -8, 2, -7, 3, -9, 4, -1, 5, -11, 6, -10, 7, -2, 8, -4, 9, -3, 10, -5, 11, -6
Dowker-Thistlethwaite code 8 14 18 16 20 22 4 2 6 12 10
A Braid Representative
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A Morse Link Presentation K11a361 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3,4\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a361/ThurstonBennequinNumber
Hyperbolic Volume 12.22
A-Polynomial See Data:K11a361/A-polynomial

[edit Notes for K11a361's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a361's four dimensional invariants]

Polynomial invariants

Alexander polynomial 7 t^2-17 t+21-17 t^{-1} +7 t^{-2}
Conway polynomial 7 z^4+11 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 69, 4 }
Jones polynomial -q^{13}+2 q^{12}-5 q^{11}+7 q^{10}-9 q^9+11 q^8-10 q^7+10 q^6-7 q^5+4 q^4-2 q^3+q^2
HOMFLY-PT polynomial (db, data sources) z^4 a^{-4} +2 z^4 a^{-6} +3 z^4 a^{-8} +z^4 a^{-10} +2 z^2 a^{-4} +3 z^2 a^{-6} +7 z^2 a^{-8} -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} +3 z^9 a^{-9} +5 z^9 a^{-11} +2 z^9 a^{-13} +5 z^8 a^{-8} +3 z^8 a^{-10} +2 z^8 a^{-14} +4 z^7 a^{-7} -7 z^7 a^{-9} -18 z^7 a^{-11} -6 z^7 a^{-13} +z^7 a^{-15} +3 z^6 a^{-6} -18 z^6 a^{-8} -20 z^6 a^{-10} -7 z^6 a^{-12} -8 z^6 a^{-14} +2 z^5 a^{-5} -9 z^5 a^{-7} +5 z^5 a^{-9} +24 z^5 a^{-11} +3 z^5 a^{-13} -5 z^5 a^{-15} +z^4 a^{-4} -5 z^4 a^{-6} +31 z^4 a^{-8} +32 z^4 a^{-10} +3 z^4 a^{-12} +8 z^4 a^{-14} -3 z^3 a^{-5} +9 z^3 a^{-7} +z^3 a^{-9} -21 z^3 a^{-11} -2 z^3 a^{-13} +8 z^3 a^{-15} -2 z^2 a^{-4} +3 z^2 a^{-6} -19 z^2 a^{-8} -21 z^2 a^{-10} +2 z^2 a^{-12} -z^2 a^{-14} +7 z a^{-11} +3 z a^{-13} -4 z a^{-15} +4 a^{-8} +2 a^{-10} - a^{-12}
The A2 invariant Data:K11a361/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a361/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{7210}{3} \frac{1262}{3} 12320 \frac{67216}{3} \frac{11968}{3} 3448 \frac{42592}{3} 39200 \frac{317240}{3} \frac{55528}{3} \frac{6336101}{30} -\frac{11042}{15} \frac{4172242}{45} \frac{27259}{18} \frac{388901}{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=4 is the signature of K11a361. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
27           1-1
25          1 1
23         41 -3
21        31  2
19       64   -2
17      53    2
15     56     1
13    55      0
11   25       3
9  25        -3
7  2         2
512          -1
31           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
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}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=8 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=9 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=10 {\mathbb Z}\oplus{\mathbb Z}_2 {\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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K11a360.gif

K11a360

K11a362.gif

K11a362