K11a60

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K11a59.gif

K11a59

K11a61.gif

K11a61

Contents

K11a60.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a60 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8394 X16,6,17,5 X10,8,11,7 X2,9,3,10 X18,12,19,11 X20,14,21,13 X6,16,7,15 X22,18,1,17 X12,20,13,19 X14,22,15,21
Gauss code 1, -5, 2, -1, 3, -8, 4, -2, 5, -4, 6, -10, 7, -11, 8, -3, 9, -6, 10, -7, 11, -9
Dowker-Thistlethwaite code 4 8 16 10 2 18 20 6 22 12 14
A Braid Representative
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A Morse Link Presentation K11a60 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a60/ThurstonBennequinNumber
Hyperbolic Volume 13.059
A-Polynomial See Data:K11a60/A-polynomial

[edit Notes for K11a60's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a60's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^3+12 t^2-18 t+19-18 t^{-1} +12 t^{-2} -3 t^{-3}
Conway polynomial -3 z^6-6 z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 85, 4 }
Jones polynomial -q^{11}+3 q^{10}-6 q^9+9 q^8-12 q^7+14 q^6-13 q^5+11 q^4-8 q^3+5 q^2-2 q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} -2 z^6 a^{-6} +z^4 a^{-2} -2 z^4 a^{-4} -8 z^4 a^{-6} +3 z^4 a^{-8} +3 z^2 a^{-2} +2 z^2 a^{-4} -10 z^2 a^{-6} +9 z^2 a^{-8} -z^2 a^{-10} + a^{-2} +2 a^{-4} -5 a^{-6} +5 a^{-8} -2 a^{-10}
Kauffman polynomial (db, data sources) z^{10} a^{-6} +z^{10} a^{-8} +3 z^9 a^{-5} +6 z^9 a^{-7} +3 z^9 a^{-9} +3 z^8 a^{-4} +4 z^8 a^{-6} +5 z^8 a^{-8} +4 z^8 a^{-10} +2 z^7 a^{-3} -9 z^7 a^{-5} -19 z^7 a^{-7} -4 z^7 a^{-9} +4 z^7 a^{-11} +z^6 a^{-2} -9 z^6 a^{-4} -23 z^6 a^{-6} -22 z^6 a^{-8} -6 z^6 a^{-10} +3 z^6 a^{-12} -6 z^5 a^{-3} +11 z^5 a^{-5} +27 z^5 a^{-7} +3 z^5 a^{-9} -6 z^5 a^{-11} +z^5 a^{-13} -4 z^4 a^{-2} +8 z^4 a^{-4} +39 z^4 a^{-6} +37 z^4 a^{-8} +4 z^4 a^{-10} -6 z^4 a^{-12} +3 z^3 a^{-3} -8 z^3 a^{-5} -14 z^3 a^{-7} -z^3 a^{-9} -2 z^3 a^{-13} +4 z^2 a^{-2} -6 z^2 a^{-4} -25 z^2 a^{-6} -22 z^2 a^{-8} -5 z^2 a^{-10} +2 z^2 a^{-12} +z a^{-5} +3 z a^{-7} +z a^{-9} +z a^{-13} - a^{-2} +2 a^{-4} +5 a^{-6} +5 a^{-8} +2 a^{-10}
The A2 invariant Data:K11a60/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a60/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a220,}

Vassiliev invariants

V2 and V3: (3, 8)
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
12 64 72 382 82 768 \frac{6592}{3} \frac{1216}{3} 416 288 2048 4584 984 \frac{126511}{10} -\frac{7706}{15} \frac{95342}{15} \frac{1073}{6} \frac{8911}{10}

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=4 is the signature of K11a60. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
23           1-1
21          2 2
19         41 -3
17        52  3
15       74   -3
13      75    2
11     67     1
9    57      -2
7   36       3
5  25        -3
3 14         3
1 1          -1
-11           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=9 {\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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K11a59.gif

K11a59

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K11a61