K11n126

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K11n125.gif

K11n125

K11n127.gif

K11n127

Contents

K11n126.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n126 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X5,19,6,18 X7,13,8,12 X2,10,3,9 X11,17,12,16 X13,21,14,20 X15,9,16,8 X17,1,18,22 X19,15,20,14 X21,7,22,6
Gauss code 1, -5, 2, -1, -3, 11, -4, 8, 5, -2, -6, 4, -7, 10, -8, 6, -9, 3, -10, 7, -11, 9
Dowker-Thistlethwaite code 4 10 -18 -12 2 -16 -20 -8 -22 -14 -6
A Braid Representative
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A Morse Link Presentation K11n126 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n126/ThurstonBennequinNumber
Hyperbolic Volume 10.7639
A-Polynomial See Data:K11n126/A-polynomial

[edit Notes for K11n126's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -6

[edit Notes for K11n126's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^3-6 t^2+4 t-1+4 t^{-1} -6 t^{-2} +3 t^{-3}
Conway polynomial 3 z^6+12 z^4+7 z^2+1
2nd Alexander ideal (db, data sources) \{3,t+1\}
Determinant and Signature { 27, 6 }
Jones polynomial q^{11}-3 q^{10}+3 q^9-5 q^8+5 q^7-3 q^6+4 q^5-2 q^4+q^3
HOMFLY-PT polynomial (db, data sources) z^6 a^{-6} +2 z^6 a^{-8} +4 z^4 a^{-6} +10 z^4 a^{-8} -2 z^4 a^{-10} +3 z^2 a^{-6} +13 z^2 a^{-8} -9 z^2 a^{-10} + a^{-6} +5 a^{-8} -7 a^{-10} +2 a^{-12}
Kauffman polynomial (db, data sources) z^9 a^{-9} +z^9 a^{-11} +3 z^8 a^{-8} +4 z^8 a^{-10} +z^8 a^{-12} +2 z^7 a^{-7} -z^7 a^{-9} -3 z^7 a^{-11} +z^6 a^{-6} -14 z^6 a^{-8} -20 z^6 a^{-10} -5 z^6 a^{-12} -7 z^5 a^{-7} -11 z^5 a^{-9} -4 z^5 a^{-11} -4 z^4 a^{-6} +19 z^4 a^{-8} +31 z^4 a^{-10} +8 z^4 a^{-12} +3 z^3 a^{-7} +20 z^3 a^{-9} +19 z^3 a^{-11} +2 z^3 a^{-13} +3 z^2 a^{-6} -13 z^2 a^{-8} -19 z^2 a^{-10} -3 z^2 a^{-12} -10 z a^{-9} -13 z a^{-11} -3 z a^{-13} - a^{-6} +5 a^{-8} +7 a^{-10} +2 a^{-12}
The A2 invariant Data:K11n126/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n126/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: (7, 16)
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
28 128 392 \frac{2306}{3} \frac{286}{3} 3584 \frac{15584}{3} \frac{2528}{3} 512 \frac{10976}{3} 8192 \frac{64568}{3} \frac{8008}{3} \frac{1096537}{30} \frac{40606}{15} \frac{480434}{45} \frac{4295}{18} \frac{37177}{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=6 is the signature of K11n126. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
012345678χ
23        11
21       2 -2
19      11 0
17     42  -2
15    22   0
13   24    2
11  221    1
9  2      2
712       -1
51        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5 i=7
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}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z} {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=8 {\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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K11n125.gif

K11n125

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K11n127