K11a55

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K11a54.gif

K11a54

K11a56.gif

K11a56

Contents

K11a55.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a55 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X16,6,17,5 X2837 X18,9,19,10 X20,11,21,12 X22,13,1,14 X6,16,7,15 X14,17,15,18 X10,19,11,20 X12,21,13,22
Gauss code 1, -4, 2, -1, 3, -8, 4, -2, 5, -10, 6, -11, 7, -9, 8, -3, 9, -5, 10, -6, 11, -7
Dowker-Thistlethwaite code 4 8 16 2 18 20 22 6 14 10 12
A Braid Representative
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BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation K11a55 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a55/ThurstonBennequinNumber
Hyperbolic Volume 11.6846
A-Polynomial See Data:K11a55/A-polynomial

[edit Notes for K11a55's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a55's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+5 t^3-10 t^2+13 t-13+13 t^{-1} -10 t^{-2} +5 t^{-3} - t^{-4}
Conway polynomial -z^8-3 z^6+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 71, -2 }
Jones polynomial -q^4+2 q^3-4 q^2+7 q-8+11 q^{-1} -11 q^{-2} +10 q^{-3} -8 q^{-4} +5 q^{-5} -3 q^{-6} + q^{-7}
HOMFLY-PT polynomial (db, data sources) -a^2 z^8+a^4 z^6-6 a^2 z^6+2 z^6+4 a^4 z^4-13 a^2 z^4-z^4 a^{-2} +10 z^4+4 a^4 z^2-13 a^2 z^2-4 z^2 a^{-2} +15 z^2+a^4-5 a^2-3 a^{-2} +8
Kauffman polynomial (db, data sources) a^2 z^{10}+z^{10}+3 a^3 z^9+5 a z^9+2 z^9 a^{-1} +4 a^4 z^8+3 a^2 z^8+2 z^8 a^{-2} +z^8+4 a^5 z^7-6 a^3 z^7-17 a z^7-6 z^7 a^{-1} +z^7 a^{-3} +4 a^6 z^6-7 a^4 z^6-19 a^2 z^6-9 z^6 a^{-2} -17 z^6+3 a^7 z^5-3 a^5 z^5+4 a^3 z^5+14 a z^5-z^5 a^{-1} -5 z^5 a^{-3} +a^8 z^4-4 a^6 z^4+7 a^4 z^4+29 a^2 z^4+12 z^4 a^{-2} +29 z^4-4 a^7 z^3-2 a^5 z^3+a^3 z^3+2 a z^3+10 z^3 a^{-1} +7 z^3 a^{-3} -a^8 z^2-4 a^4 z^2-19 a^2 z^2-8 z^2 a^{-2} -22 z^2+a^7 z+2 a^5 z-a^3 z-5 a z-6 z a^{-1} -3 z a^{-3} +a^4+5 a^2+3 a^{-2} +8
The A2 invariant q^{20}-q^{18}+q^{16}-q^{14}-q^{12}-3 q^8+2 q^6-q^4+3 q^2+3+ q^{-2} +2 q^{-4} - q^{-6} - q^{-10} - q^{-12}
The G2 invariant q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+4 q^{106}-2 q^{104}-4 q^{102}+12 q^{100}-18 q^{98}+22 q^{96}-19 q^{94}+8 q^{92}+7 q^{90}-23 q^{88}+37 q^{86}-40 q^{84}+34 q^{82}-19 q^{80}-2 q^{78}+23 q^{76}-36 q^{74}+46 q^{72}-44 q^{70}+33 q^{68}-16 q^{66}-8 q^{64}+30 q^{62}-44 q^{60}+48 q^{58}-35 q^{56}+11 q^{54}+15 q^{52}-37 q^{50}+35 q^{48}-16 q^{46}-16 q^{44}+41 q^{42}-49 q^{40}+26 q^{38}+11 q^{36}-55 q^{34}+81 q^{32}-86 q^{30}+51 q^{28}-2 q^{26}-53 q^{24}+93 q^{22}-101 q^{20}+80 q^{18}-33 q^{16}-17 q^{14}+59 q^{12}-77 q^{10}+73 q^8-34 q^6-7 q^4+46 q^2-52+39 q^{-2} +6 q^{-4} -44 q^{-6} +68 q^{-8} -59 q^{-10} +26 q^{-12} +25 q^{-14} -68 q^{-16} +92 q^{-18} -78 q^{-20} +42 q^{-22} +3 q^{-24} -47 q^{-26} +66 q^{-28} -65 q^{-30} +45 q^{-32} -19 q^{-34} -7 q^{-36} +22 q^{-38} -29 q^{-40} +23 q^{-42} -16 q^{-44} +6 q^{-46} -5 q^{-50} +4 q^{-52} -4 q^{-54} +3 q^{-56} - q^{-58} + q^{-60}

"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{124}{3} \frac{44}{3} 64 \frac{176}{3} -\frac{160}{3} 40 \frac{256}{3} 32 \frac{992}{3} \frac{352}{3} \frac{7231}{15} \frac{292}{5} \frac{10324}{45} \frac{353}{9} \frac{271}{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=-2 is the signature of K11a55. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012345χ
9           1-1
7          1 1
5         31 -2
3        41  3
1       43   -1
-1      74    3
-3     55     0
-5    56      -1
-7   35       2
-9  25        -3
-11 13         2
-13 2          -2
-151           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\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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K11a54.gif

K11a54

K11a56.gif

K11a56