K11a346

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K11a345.gif

K11a345

K11a347.gif

K11a347

Contents

K11a346.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a346 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X16,3,17,4 X12,6,13,5 X18,8,19,7 X20,10,21,9 X4,12,5,11 X22,13,1,14 X10,16,11,15 X2,17,3,18 X8,20,9,19 X14,21,15,22
Gauss code 1, -9, 2, -6, 3, -1, 4, -10, 5, -8, 6, -3, 7, -11, 8, -2, 9, -4, 10, -5, 11, -7
Dowker-Thistlethwaite code 6 16 12 18 20 4 22 10 2 8 14
A Braid Representative
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A Morse Link Presentation K11a346 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a346/ThurstonBennequinNumber
Hyperbolic Volume 14.6898
A-Polynomial See Data:K11a346/A-polynomial

[edit Notes for K11a346's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a346's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+12 t^2-18 t+21-18 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6+2 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 93, 4 }
Jones polynomial -q^9+3 q^8-6 q^7+10 q^6-13 q^5+14 q^4-14 q^3+13 q^2-9 q+6-3 q^{-1} + q^{-2}
HOMFLY-PT polynomial (db, data sources) z^8 a^{-4} -2 z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -9 z^4 a^{-2} +14 z^4 a^{-4} -4 z^4 a^{-6} +z^4-12 z^2 a^{-2} +15 z^2 a^{-4} -5 z^2 a^{-6} +3 z^2-4 a^{-2} +5 a^{-4} -2 a^{-6} +2
Kauffman polynomial (db, data sources) 2 z^{10} a^{-2} +2 z^{10} a^{-4} +3 z^9 a^{-1} +10 z^9 a^{-3} +7 z^9 a^{-5} -3 z^8 a^{-2} +8 z^8 a^{-4} +12 z^8 a^{-6} +z^8-15 z^7 a^{-1} -40 z^7 a^{-3} -12 z^7 a^{-5} +13 z^7 a^{-7} -17 z^6 a^{-2} -53 z^6 a^{-4} -31 z^6 a^{-6} +10 z^6 a^{-8} -5 z^6+24 z^5 a^{-1} +42 z^5 a^{-3} -17 z^5 a^{-5} -29 z^5 a^{-7} +6 z^5 a^{-9} +41 z^4 a^{-2} +68 z^4 a^{-4} +19 z^4 a^{-6} -14 z^4 a^{-8} +3 z^4 a^{-10} +9 z^4-12 z^3 a^{-1} -6 z^3 a^{-3} +27 z^3 a^{-5} +17 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} -26 z^2 a^{-2} -30 z^2 a^{-4} -7 z^2 a^{-6} +4 z^2 a^{-8} -7 z^2-3 z a^{-3} -7 z a^{-5} -4 z a^{-7} +4 a^{-2} +5 a^{-4} +2 a^{-6} +2
The A2 invariant q^6+1-2 q^{-2} +2 q^{-4} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} + q^{-20} -2 q^{-22} + q^{-24} - q^{-26}
The G2 invariant Data:K11a346/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a106, K11a194,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {}

Vassiliev invariants

V2 and V3: (1, 3)
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
4 24 8 \frac{158}{3} -\frac{38}{3} 96 176 -128 120 \frac{32}{3} 288 \frac{632}{3} -\frac{152}{3} \frac{25711}{30} -\frac{3634}{5} \frac{39302}{45} \frac{1265}{18} \frac{3631}{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 K11a346. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
19           1-1
17          2 2
15         41 -3
13        62  4
11       74   -3
9      76    1
7     77     0
5    67      -1
3   48       4
1  25        -3
-1 14         3
-3 2          -2
-51           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\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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K11a345.gif

K11a345

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K11a347