K11a10

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K11a9.gif

K11a9

K11a11.gif

K11a11

Contents

K11a10.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a10 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X16,7,17,8 X2,9,3,10 X18,12,19,11 X20,14,21,13 X22,16,1,15 X6,17,7,18 X14,20,15,19 X12,22,13,21
Gauss code 1, -5, 2, -1, 3, -9, 4, -2, 5, -3, 6, -11, 7, -10, 8, -4, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 8 10 16 2 18 20 22 6 14 12
A Braid Representative
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A Morse Link Presentation K11a10 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:K11a10/ThurstonBennequinNumber
Hyperbolic Volume 14.6473
A-Polynomial See Data:K11a10/A-polynomial

[edit Notes for K11a10'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 K11a10's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-11 t^2+25 t-31+25 t^{-1} -11 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 107, 2 }
Jones polynomial -q^8+3 q^7-6 q^6+11 q^5-15 q^4+17 q^3-17 q^2+15 q-11+7 q^{-1} -3 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2-2 z^2 a^{-2} +5 z^2 a^{-4} -2 z^2 a^{-6} -3 z^2+a^2-2 a^{-2} +3 a^{-4} - a^{-6}
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +3 z^9 a^{-1} +7 z^9 a^{-3} +4 z^9 a^{-5} +8 z^8 a^{-2} +10 z^8 a^{-4} +6 z^8 a^{-6} +4 z^8+3 a z^7-11 z^7 a^{-3} -3 z^7 a^{-5} +5 z^7 a^{-7} +a^2 z^6-22 z^6 a^{-2} -30 z^6 a^{-4} -13 z^6 a^{-6} +3 z^6 a^{-8} -7 z^6-8 a z^5-11 z^5 a^{-1} +4 z^5 a^{-3} -4 z^5 a^{-5} -10 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+19 z^4 a^{-2} +37 z^4 a^{-4} +14 z^4 a^{-6} -6 z^4 a^{-8} -z^4+6 a z^3+6 z^3 a^{-1} +2 z^3 a^{-3} +11 z^3 a^{-5} +7 z^3 a^{-7} -2 z^3 a^{-9} +3 a^2 z^2-11 z^2 a^{-2} -18 z^2 a^{-4} -5 z^2 a^{-6} +2 z^2 a^{-8} +3 z^2-a z-z a^{-1} -2 z a^{-3} -4 z a^{-5} -2 z a^{-7} -a^2+2 a^{-2} +3 a^{-4} + a^{-6}
The A2 invariant Data:K11a10/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a10/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a262,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a42,}

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{208}{3} -\frac{256}{3} 8 -\frac{32}{3} 32 -\frac{328}{3} \frac{40}{3} -\frac{6271}{30} -\frac{4858}{15} \frac{7738}{45} -\frac{449}{18} \frac{1889}{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=2 is the signature of K11a10. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
17           1-1
15          2 2
13         41 -3
11        72  5
9       84   -4
7      97    2
5     88     0
3    79      -2
1   59       4
-1  26        -4
-3 15         4
-5 2          -2
-71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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K11a9

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K11a11