K11a50

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K11a49.gif

K11a49

K11a51.gif

K11a51

Contents

K11a50.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a50 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8394 X14,6,15,5 X10,8,11,7 X2,9,3,10 X18,12,19,11 X6,14,7,13 X22,16,1,15 X20,18,21,17 X12,20,13,19 X16,22,17,21
Gauss code 1, -5, 2, -1, 3, -7, 4, -2, 5, -4, 6, -10, 7, -3, 8, -11, 9, -6, 10, -9, 11, -8
Dowker-Thistlethwaite code 4 8 14 10 2 18 6 22 20 12 16
A Braid Representative
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A Morse Link Presentation K11a50 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a50/ThurstonBennequinNumber
Hyperbolic Volume 13.4737
A-Polynomial See Data:K11a50/A-polynomial

[edit Notes for K11a50's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a50's four dimensional invariants]

Polynomial invariants

Alexander polynomial -5 t^2+21 t-31+21 t^{-1} -5 t^{-2}
Conway polynomial -5 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 83, 2 }
Jones polynomial -q^{10}+3 q^9-6 q^8+9 q^7-11 q^6+13 q^5-13 q^4+11 q^3-8 q^2+5 q-2+ q^{-1}
HOMFLY-PT polynomial (db, data sources) -z^4 a^{-2} -2 z^4 a^{-4} -2 z^4 a^{-6} -z^2 a^{-4} -2 z^2 a^{-6} +3 z^2 a^{-8} +z^2- a^{-6} +2 a^{-8} - a^{-10} +1
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} +4 z^8 a^{-4} +6 z^8 a^{-6} +5 z^8 a^{-8} +3 z^8 a^{-10} +4 z^7 a^{-3} -2 z^7 a^{-5} -15 z^7 a^{-7} -8 z^7 a^{-9} +z^7 a^{-11} +3 z^6 a^{-2} -5 z^6 a^{-4} -24 z^6 a^{-6} -28 z^6 a^{-8} -12 z^6 a^{-10} +2 z^5 a^{-1} -5 z^5 a^{-3} -7 z^5 a^{-5} +3 z^5 a^{-7} -z^5 a^{-9} -4 z^5 a^{-11} -2 z^4 a^{-2} +2 z^4 a^{-4} +23 z^4 a^{-6} +32 z^4 a^{-8} +14 z^4 a^{-10} +z^4-2 z^3 a^{-1} +5 z^3 a^{-3} +10 z^3 a^{-5} +7 z^3 a^{-7} +9 z^3 a^{-9} +5 z^3 a^{-11} +z^2 a^{-4} -8 z^2 a^{-6} -13 z^2 a^{-8} -6 z^2 a^{-10} -2 z^2-2 z a^{-3} -4 z a^{-5} -3 z a^{-7} -3 z a^{-9} -2 z a^{-11} + a^{-6} +2 a^{-8} + a^{-10} +1
The A2 invariant Data:K11a50/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a50/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: (1, 4)
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 32 8 \frac{542}{3} \frac{130}{3} 128 \frac{2432}{3} \frac{416}{3} 192 \frac{32}{3} 512 \frac{2168}{3} \frac{520}{3} \frac{106831}{30} -\frac{5182}{15} \frac{90302}{45} \frac{2225}{18} \frac{7951}{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 K11a50. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
21           1-1
19          2 2
17         41 -3
15        52  3
13       64   -2
11      75    2
9     66     0
7    57      -2
5   36       3
3  25        -3
1 14         3
-1 1          -1
-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}\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}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
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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K11a49.gif

K11a49

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K11a51