K11n125

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K11n124.gif

K11n124

K11n126.gif

K11n126

Contents

K11n125.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n125 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X5,17,6,16 X7,19,8,18 X14,10,15,9 X2,11,3,12 X20,14,21,13 X22,15,1,16 X17,7,18,6 X12,20,13,19 X8,21,9,22
Gauss code 1, -6, 2, -1, -3, 9, -4, -11, 5, -2, 6, -10, 7, -5, 8, 3, -9, 4, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 10 -16 -18 14 2 20 22 -6 12 8
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation K11n125 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n125/ThurstonBennequinNumber
Hyperbolic Volume 14.0835
A-Polynomial See Data:K11n125/A-polynomial

[edit Notes for K11n125's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -2

[edit Notes for K11n125's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-6 t^2+15 t-19+15 t^{-1} -6 t^{-2} + t^{-3}
Conway polynomial z^6+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 63, 2 }
Jones polynomial -q^8+4 q^7-7 q^6+9 q^5-11 q^4+11 q^3-9 q^2+7 q-3+ q^{-1}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-4} -2 z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -3 z^2 a^{-2} +3 z^2 a^{-4} -z^2 a^{-6} +z^2+1
Kauffman polynomial (db, data sources) z^9 a^{-3} +z^9 a^{-5} +z^8 a^{-2} +5 z^8 a^{-4} +4 z^8 a^{-6} +6 z^7 a^{-5} +6 z^7 a^{-7} -z^6 a^{-2} -9 z^6 a^{-4} -4 z^6 a^{-6} +4 z^6 a^{-8} +3 z^5 a^{-1} +2 z^5 a^{-3} -14 z^5 a^{-5} -12 z^5 a^{-7} +z^5 a^{-9} +6 z^4 a^{-2} +9 z^4 a^{-4} -3 z^4 a^{-6} -7 z^4 a^{-8} +z^4-3 z^3 a^{-1} +8 z^3 a^{-5} +4 z^3 a^{-7} -z^3 a^{-9} -5 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} +z^2 a^{-8} -2 z^2+1
The A2 invariant q^4-1+3 q^{-2} - q^{-4} +2 q^{-6} + q^{-8} -2 q^{-10} + q^{-12} -3 q^{-14} +2 q^{-16} - q^{-20} +2 q^{-22} - q^{-24}
The G2 invariant Data:K11n125/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n176,}

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

Vassiliev invariants

V2 and V3: (0, 0)
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
0 0 0 0 0 0 0 -32 32 0 0 0 0 0 -192 192 0 32

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=2 is the signature of K11n125. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-101234567χ
17         1-1
15        3 3
13       41 -3
11      53  2
9     64   -2
7    55    0
5   46     2
3  35      -2
1 15       4
-1 2        -2
-31         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11n124.gif

K11n124

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K11n126