K11n12

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K11n11.gif

K11n11

K11n13.gif

K11n13

Contents

K11n12.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n12 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X7,14,8,15 X2,9,3,10 X11,19,12,18 X13,6,14,7 X15,1,16,22 X17,21,18,20 X19,13,20,12 X21,17,22,16
Gauss code 1, -5, 2, -1, 3, 7, -4, -2, 5, -3, -6, 10, -7, 4, -8, 11, -9, 6, -10, 9, -11, 8
Dowker-Thistlethwaite code 4 8 10 -14 2 -18 -6 -22 -20 -12 -16
A Braid Representative
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A Morse Link Presentation K11n12 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:K11n12/ThurstonBennequinNumber
Hyperbolic Volume 8.07031
A-Polynomial See Data:K11n12/A-polynomial

[edit Notes for K11n12's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n12's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^2-3 t+5-3 t^{-1} + t^{-2}
Conway polynomial z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 13, 0 }
Jones polynomial -q^7+2 q^6-2 q^5+2 q^4-2 q^3+2 q^2-q+1
HOMFLY-PT polynomial (db, data sources) z^4 a^{-4} -z^2 a^{-2} +3 z^2 a^{-4} -z^2 a^{-6} - a^{-2} +2 a^{-4} - a^{-6} +1
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} -6 z^6 a^{-2} -17 z^6 a^{-4} -11 z^6 a^{-6} +5 z^5 a^{-3} -5 z^5 a^{-7} +10 z^4 a^{-2} +26 z^4 a^{-4} +16 z^4 a^{-6} -z^3 a^{-1} +7 z^3 a^{-5} +6 z^3 a^{-7} -7 z^2 a^{-2} -13 z^2 a^{-4} -7 z^2 a^{-6} -z^2-z a^{-3} -3 z a^{-5} -2 z a^{-7} + a^{-2} +2 a^{-4} + a^{-6} +1
The A2 invariant Data:K11n12/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n12/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {6_3,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {9_43,}

Vassiliev invariants

V2 and V3: (1, 2)
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 16 8 \frac{110}{3} -\frac{14}{3} 64 \frac{352}{3} -\frac{32}{3} 16 \frac{32}{3} 128 \frac{440}{3} -\frac{56}{3} \frac{13471}{30} \frac{338}{15} \frac{3902}{45} \frac{65}{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 K11n12. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567χ
15       1-1
13      1 1
11     11 0
9   121  0
7   11   0
5  22    0
3111     1
111      0
-11       1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1 i=3
r=0 {\mathbb Z} {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}\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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K11n11.gif

K11n11

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K11n13