K11n28

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K11n27.gif

K11n27

K11n29.gif

K11n29

Contents

K11n28.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n28 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X12,5,13,6 X2837 X9,15,10,14 X11,18,12,19 X6,13,7,14 X15,1,16,22 X17,21,18,20 X19,10,20,11 X21,17,22,16
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, -5, 10, -6, -3, 7, 5, -8, 11, -9, 6, -10, 9, -11, 8
Dowker-Thistlethwaite code 4 8 12 2 -14 -18 6 -22 -20 -10 -16
A Braid Representative
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A Morse Link Presentation K11n28 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n28/ThurstonBennequinNumber
Hyperbolic Volume 8.06165
A-Polynomial See Data:K11n28/A-polynomial

[edit Notes for K11n28's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 2
Rasmussen s-Invariant 0

[edit Notes for K11n28's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^2-5 t+9-5 t^{-1} + t^{-2}
Conway polynomial z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 21, 0 }
Jones polynomial -q^7+2 q^6-2 q^5+3 q^4-3 q^3+3 q^2-3 q+2- q^{-1} + q^{-2}
HOMFLY-PT polynomial (db, data sources) z^4 a^{-4} -2 z^2 a^{-2} +3 z^2 a^{-4} -z^2 a^{-6} -z^2+a^2-2 a^{-2} +3 a^{-4} - a^{-6}
Kauffman polynomial (db, data sources) z^9 a^{-3} +z^9 a^{-5} +z^8 a^{-2} +3 z^8 a^{-4} +2 z^8 a^{-6} -5 z^7 a^{-3} -4 z^7 a^{-5} +z^7 a^{-7} -5 z^6 a^{-2} -16 z^6 a^{-4} -11 z^6 a^{-6} +2 z^5 a^{-1} +9 z^5 a^{-3} +2 z^5 a^{-5} -5 z^5 a^{-7} +10 z^4 a^{-2} +26 z^4 a^{-4} +17 z^4 a^{-6} +z^4+a z^3-5 z^3 a^{-1} -10 z^3 a^{-3} +2 z^3 a^{-5} +6 z^3 a^{-7} +a^2 z^2-10 z^2 a^{-2} -17 z^2 a^{-4} -8 z^2 a^{-6} +3 z a^{-1} +4 z a^{-3} -z a^{-7} -a^2+2 a^{-2} +3 a^{-4} + a^{-6}
The A2 invariant Data:K11n28/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n28/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {7_7,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n64,}

Vassiliev invariants

V2 and V3: (-1, 1)
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 8 8 \frac{82}{3} -\frac{10}{3} -32 -\frac{112}{3} -\frac{160}{3} 8 -\frac{32}{3} 32 -\frac{328}{3} \frac{40}{3} -\frac{3391}{30} -\frac{2218}{15} \frac{3418}{45} -\frac{161}{18} \frac{449}{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=0 is the signature of K11n28. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-101234567χ
15         1-1
13        1 1
11       11 0
9      21  1
7     11   0
5    22    0
3   11     0
1  12      -1
-1 12       1
-3          0
-51         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{2} {\mathbb Z}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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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K11n27.gif

K11n27

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K11n29