K11a49

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K11a48.gif

K11a48

K11a50.gif

K11a50

Contents

K11a49.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a49 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 X20,16,21,15 X22,18,1,17 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, -8, 11, -9
Dowker-Thistlethwaite code 4 8 14 10 2 18 6 20 22 12 16
A Braid Representative
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A Morse Link Presentation K11a49 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{2,3\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a49/ThurstonBennequinNumber
Hyperbolic Volume 15.1714
A-Polynomial See Data:K11a49/A-polynomial

[edit Notes for K11a49'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 K11a49's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^3+13 t^2-23 t+27-23 t^{-1} +13 t^{-2} -3 t^{-3}
Conway polynomial -3 z^6-5 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 105, 4 }
Jones polynomial -q^{11}+3 q^{10}-7 q^9+12 q^8-15 q^7+17 q^6-17 q^5+14 q^4-10 q^3+6 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} -7 z^4 a^{-6} +3 z^4 a^{-8} +3 z^2 a^{-2} -8 z^2 a^{-6} +8 z^2 a^{-8} -z^2 a^{-10} +2 a^{-2} -4 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} +7 z^9 a^{-7} +4 z^9 a^{-9} +3 z^8 a^{-4} +7 z^8 a^{-6} +10 z^8 a^{-8} +6 z^8 a^{-10} +2 z^7 a^{-3} -5 z^7 a^{-5} -14 z^7 a^{-7} -2 z^7 a^{-9} +5 z^7 a^{-11} +z^6 a^{-2} -6 z^6 a^{-4} -24 z^6 a^{-6} -30 z^6 a^{-8} -10 z^6 a^{-10} +3 z^6 a^{-12} -5 z^5 a^{-3} +5 z^5 a^{-5} +16 z^5 a^{-7} -2 z^5 a^{-9} -7 z^5 a^{-11} +z^5 a^{-13} -4 z^4 a^{-2} +2 z^4 a^{-4} +32 z^4 a^{-6} +41 z^4 a^{-8} +10 z^4 a^{-10} -5 z^4 a^{-12} +2 z^3 a^{-3} -8 z^3 a^{-5} -13 z^3 a^{-7} +z^3 a^{-9} +2 z^3 a^{-11} -2 z^3 a^{-13} +5 z^2 a^{-2} -21 z^2 a^{-6} -26 z^2 a^{-8} -8 z^2 a^{-10} +2 z^2 a^{-12} +z a^{-3} +4 z a^{-5} +5 z a^{-7} +z a^{-9} +z a^{-13} -2 a^{-2} +4 a^{-6} +5 a^{-8} +2 a^{-10}
The A2 invariant Data:K11a49/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a49/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, 6)
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 48 32 \frac{844}{3} \frac{164}{3} 384 1440 256 240 \frac{256}{3} 1152 \frac{6752}{3} \frac{1312}{3} \frac{111511}{15} \frac{116}{15} \frac{144364}{45} \frac{1097}{9} \frac{5911}{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=4 is the signature of K11a49. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
23           1-1
21          2 2
19         51 -4
17        72  5
15       85   -3
13      97    2
11     88     0
9    69      -3
7   48       4
5  26        -4
3 15         4
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}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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K11a48

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K11a50