K11a328

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K11a327.gif

K11a327

K11a329.gif

K11a329

Contents

K11a328.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a328 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X12,4,13,3 X18,6,19,5 X16,7,17,8 X22,10,1,9 X4,12,5,11 X20,14,21,13 X8,15,9,16 X2,18,3,17 X10,20,11,19 X14,22,15,21
Gauss code 1, -9, 2, -6, 3, -1, 4, -8, 5, -10, 6, -2, 7, -11, 8, -4, 9, -3, 10, -7, 11, -5
Dowker-Thistlethwaite code 6 12 18 16 22 4 20 8 2 10 14
A Braid Representative
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A Morse Link Presentation K11a328 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a328/ThurstonBennequinNumber
Hyperbolic Volume 17.6919
A-Polynomial See Data:K11a328/A-polynomial

[edit Notes for K11a328's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -4

[edit Notes for K11a328's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^3+16 t^2-34 t+43-34 t^{-1} +16 t^{-2} -3 t^{-3}
Conway polynomial -3 z^6-2 z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 149, 4 }
Jones polynomial -q^{11}+4 q^{10}-9 q^9+15 q^8-21 q^7+24 q^6-24 q^5+21 q^4-15 q^3+10 q^2-4 q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} -2 z^6 a^{-6} +z^4 a^{-2} -6 z^4 a^{-6} +3 z^4 a^{-8} +z^2 a^{-2} +5 z^2 a^{-4} -8 z^2 a^{-6} +6 z^2 a^{-8} -z^2 a^{-10} +4 a^{-4} -5 a^{-6} +3 a^{-8} - a^{-10}
Kauffman polynomial (db, data sources) 3 z^{10} a^{-6} +3 z^{10} a^{-8} +8 z^9 a^{-5} +16 z^9 a^{-7} +8 z^9 a^{-9} +8 z^8 a^{-4} +11 z^8 a^{-6} +13 z^8 a^{-8} +10 z^8 a^{-10} +4 z^7 a^{-3} -15 z^7 a^{-5} -33 z^7 a^{-7} -6 z^7 a^{-9} +8 z^7 a^{-11} +z^6 a^{-2} -19 z^6 a^{-4} -43 z^6 a^{-6} -42 z^6 a^{-8} -15 z^6 a^{-10} +4 z^6 a^{-12} -8 z^5 a^{-3} +6 z^5 a^{-5} +19 z^5 a^{-7} -8 z^5 a^{-9} -12 z^5 a^{-11} +z^5 a^{-13} -2 z^4 a^{-2} +15 z^4 a^{-4} +46 z^4 a^{-6} +43 z^4 a^{-8} +9 z^4 a^{-10} -5 z^4 a^{-12} +3 z^3 a^{-3} +2 z^3 a^{-7} +12 z^3 a^{-9} +6 z^3 a^{-11} -z^3 a^{-13} +z^2 a^{-2} -10 z^2 a^{-4} -23 z^2 a^{-6} -17 z^2 a^{-8} -4 z^2 a^{-10} +z^2 a^{-12} -2 z a^{-5} -3 z a^{-7} -3 z a^{-9} -2 z a^{-11} +4 a^{-4} +5 a^{-6} +3 a^{-8} + a^{-10}
The A2 invariant Data:K11a328/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a328/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: (3, 6)
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
12 48 72 254 50 576 1376 224 272 288 1152 3048 600 \frac{73711}{10} -\frac{7306}{15} \frac{18874}{5} \frac{347}{2} \frac{5231}{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=4 is the signature of K11a328. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
23           1-1
21          3 3
19         61 -5
17        93  6
15       126   -6
13      129    3
11     1212     0
9    912      -3
7   612       6
5  49        -5
3 17         6
1 3          -3
-11           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=4 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=5 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=9 {\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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K11a327.gif

K11a327

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K11a329