K11n110

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K11n109.gif

K11n109

K11n111.gif

K11n111

Contents

K11n110.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n110 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X14,6,15,5 X7,17,8,16 X9,19,10,18 X2,11,3,12 X20,13,21,14 X22,16,1,15 X17,9,18,8 X12,19,13,20 X6,21,7,22
Gauss code 1, -6, 2, -1, 3, -11, -4, 9, -5, -2, 6, -10, 7, -3, 8, 4, -9, 5, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 10 14 -16 -18 2 20 22 -8 12 6
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation K11n110 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n110/ThurstonBennequinNumber
Hyperbolic Volume 12.5494
A-Polynomial See Data:K11n110/A-polynomial

[edit Notes for K11n110's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11n110's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+4 t^2-9 t+13-9 t^{-1} +4 t^{-2} - t^{-3}
Conway polynomial -z^6-2 z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 41, 0 }
Jones polynomial q^6-3 q^5+5 q^4-6 q^3+7 q^2-7 q+6-4 q^{-1} +2 q^{-2}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -4 z^4 a^{-2} +z^4 a^{-4} +z^4-5 z^2 a^{-2} +2 z^2 a^{-4} +z^2+a^2- a^{-2} + a^{-4}
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +4 z^8 a^{-2} +3 z^8 a^{-4} +z^8-2 z^7 a^{-1} +z^7 a^{-3} +3 z^7 a^{-5} -13 z^6 a^{-2} -9 z^6 a^{-4} +z^6 a^{-6} -3 z^6+a z^5+z^5 a^{-1} -10 z^5 a^{-3} -10 z^5 a^{-5} +14 z^4 a^{-2} +5 z^4 a^{-4} -3 z^4 a^{-6} +6 z^4+2 a z^3+2 z^3 a^{-1} +7 z^3 a^{-3} +7 z^3 a^{-5} +2 a^2 z^2-7 z^2 a^{-2} -2 z^2 a^{-4} +2 z^2 a^{-6} -z^2-a z-z a^{-1} -z a^{-3} -z a^{-5} -a^2+ a^{-2} + a^{-4}
The A2 invariant q^8+2 q^6-q^4+q^2-1- q^{-2} + q^{-4} - q^{-6} +2 q^{-8} - q^{-10} + q^{-14} - q^{-16} + q^{-18}
The G2 invariant Data:K11n110/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: (-2, -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
-8 -8 32 \frac{164}{3} \frac{100}{3} 64 \frac{400}{3} \frac{160}{3} 24 -\frac{256}{3} 32 -\frac{1312}{3} -\frac{800}{3} -\frac{4111}{15} \frac{1388}{5} -\frac{26884}{45} \frac{943}{9} -\frac{2431}{15}

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 K11n110. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456χ
13        11
11       2 -2
9      31 2
7     32  -1
5    43   1
3   33    0
1  34     -1
-1 24      2
-3 2       -2
-52        2
Integral Khovanov Homology

(db, data source)

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

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K11n109.gif

K11n109

K11n111.gif

K11n111